Physics 110 Lecture 2 Summer 2024 PDF

Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 1 FORCES AND NEWTON’S LAWS Chapters 2,4,5  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 2 Student learning objectives Inertia Force Newton's First Law of Motion Net Force and Vectors T...

Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 1 FORCES AND NEWTON’S LAWS Chapters 2,4,5  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 2 Student learning objectives Inertia Force Newton's First Law of Motion Net Force and Vectors The Equilibrium Rule Support Force Equilibrium of Moving Things Force Causes Acceleration Friction Mass and Weight Newton's Second Law of Motion Free Fall Nonfree Fall Newton’s Third Law of Motion Vectors and the Third Law  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 3 Dynamics Dynamics (the study of forces: why things move) is essentially made up of Newton’s Three Laws of motion. The textbook divides these up into three chapters (2,4,5), but we’re going to compress them into one lecture. We should probably start with “What is a force?”  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 4 Inertia Objects of different weight fall to the ground at the same rate in the absence of air resistance. A moving object needs no force to keep it moving in the absence of friction. Inertia is a property of matter to resist changes in motion. It depends on the amount of matter in an object (its mass).  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 5 Forces A force is any push or pull between two objects. Notice that there must be two objects involved Some examples: Gravitational force (weight) – your book is very sloppy with this terminology. For us, the force of gravity will be synonymous with weight Tension – in ropes and strings and wires Normal force – your book calls this “support force”; it’s the force that a surface exerts to keep you from sinking into it Friction – acts opposite to relative motion due to surfaces Buoyant force – upward force in fluids Electric force  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 6 Forces Forces are vectors, since they have a magnitude and a direction. The SI unit of force is the Newton (N) (in Imperial units, it’s the pound (lb.)) When there are many forces acting on an object, we are usually interested in the total or net force, which is the (vector) sum of all the forces. ⃗ I’ll call the net force Σ끫롲 끫롲  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 7 Net Force Since it’s a vector sum, the net force depends on the magnitudes and directions of the applied forces.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 8 Equilibrium When the net force on an object is zero ⃗끫롲 = 0), we say that the object is in (Σ끫롲 equilibrium (more specifically, mechanical equilibrium). Things that are in mechanical equilibrium have no change in their motion. An unbalanced (nonzero net) force would be needed to change their state of motion  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 9 Equilibrium When the girl holds the rock with as much force upward as gravity pulls downward, the net force on the rock is zero.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 10 Equilibrium The upward tension in the string has the same magnitude as the weight of the bag, so the net force on the bag is zero. The bag of sugar is attracted to Earth with a gravitational force of 2 pounds or 9 Newtons.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 11 Equilibrium Two forces act on the bag Tension force in string acts upward. Force due to gravity acts downward. Both are equal in magnitude and opposite in direction. When added, they cancel to zero. So, the bag remains at rest.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 12 Vectors vs. Scalars Remember the difference!  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 13 Equilibrium The scaffold is suspended by two ropes attached to its left and right end. A painter stands at the left end of the scaffold, to the left of the rope. Another painter stands at the right end, just right of center, and is moving towards the left..  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 14 Equilibrium The sum of the upward vectors equals the sum of the ⃗끫롲 = 0, and the scaffold is in downward vectors. Σ끫롲 equilibrium. Two upward force vectors are drawn along the ropes, with a longer one on the left and shorter one on the right. Three downward force vectors are drawn, one at each painter and one at the center of the scaffold. F(upwards) = F(downwards) ⃗끫롲 = 0 Σ끫롲  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 15 Normal (Support) Forces In physics, the word “normal” means “perpendicular”, so the normal force of a surface on an object is the force the surface exerts perpendicular to that surface.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 16 Quick Question When you stand on two bathroom scales with one foot on each scale and with your weight evenly distributed, each scale will read A. your weight. B. half your weight. C. zero. D. more than your weight.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 17 Quick Question When you stand on two bathroom scales with one foot on each scale and with your weight evenly distributed, each scale will read A. your weight. B. half your weight. C. zero. D. more than your weight. ⃗=0 You are at rest, so Σ끫롲 끫롲 The sum of the two upward support forces is equal to your weight. Your weight is distributed over the two scales. So, each scale reads half your weight.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 18 Quick Question If the gymnast hangs with her weight evenly divided between the two rings, how would scale readings in both supporting ropes compare with her weight? Suppose she hangs with slightly more of her weight supported by the left ring. How would a scale on the right read?  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 19 Quick Question If the gymnast hangs with her weight evenly divided between the two rings, how would scale readings in both supporting ropes compare with her weight? Each half Suppose she hangs with slightly more of her weight supported by the left ring. How would a scale on the right read? Less than half  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 20 Equilibrium So far, we’ve looked at static equilibrium, where the object of interest is at rest. ⃗ = 0, since nothing is moving. It’s clear in that case that Σ끫롲 끫롲 However, we may also have dynamic equilibrium, where an object is moving at constant speed in a straight line. This type of motion is also called rectilinear motion, and it means once again that Σ끫롲 ⃗ = 0. 끫롲  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 21 Equilibrium Equilibrium test: whether something undergoes change in motion Static equilibrium: A crate at rest (no change in motion). Dynamic equilibrium: A crate pushed at steady speed (no change in motion).  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 22 Quick Question An airplane flies horizontally at constant speed in a straight-line direction. Its state of motion is unchanging. In other words, it is in equilibrium. Two horizontal forces act on the plane. One is the thrust of the propeller that pulls it forward. The other is the force of air resistance (air friction) that acts in the opposite direction. Which force is greater?  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 23 Quick Question An airplane flies horizontally at constant speed in a straight-line direction. Its state of motion is unchanging. In other words, it is in equilibrium. Two horizontal forces act on the plane. One is the thrust of the propeller that pulls it forward. The other is the force of air resistance (air friction) that acts in the opposite direction. Which force is greater? They are the same!  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 24 Friction Depends on the kinds of material and how much they are pressed together. Is due to tiny surface bumps and to "stickiness" of the atoms on a material's surface. Example: Friction between a crate on a smooth wooden floor is less than that on a rough floor.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 25 Friction Surfaces can also exert forces parallel to the surface (remember that normal forces are perpendicular). These forces are frictional forces. They come in two types, static friction and kinetic (or dynamic) friction. Static friction exists before an object moves to oppose another force, while kinetic exists when an object is sliding.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 26 Quick Question You push a crate at a steady speed in a straight line. If the friction force is 75 N, how much force must you apply? A. More than 75 N. B. Less than 75 N. C. Equal to 75 N. D. Not enough information.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 27 Quick Question You push a crate at a steady speed in a straight line. If the friction force is 75 N, how much force must you apply? A. More than 75 N. B. Less than 75 N. C. Equal to 75 N. D. Not enough information. The crate is in dynamic equilibrium. So there is no NET force acting on it. So you must apply a force that balances out the force of friction. Applied force must be equal and opposite to the force of friction. So if Friction is 75 N to the left, the Applied Force from you is 75 N to the right.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 28 Dynamic Equilibrium When the push on the desk is the same as the force of friction between the desk and the floor, the net force is zero and the desk slides at an unchanging speed.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 29 Quick Question You are riding in a vehicle at a steady speed and toss a coin straight upward. Where will the coin land? A. Behind you. B. Ahead of you. C. In your hand. D. There is not enough information.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 30 Quick Question You are riding in a vehicle at a steady speed and toss a coin straight upward. Where will the coin land? A. Behind you. B. Ahead of you. C. In your hand. D. There is not enough information. Due to the coin's inertia, it continues sideways with the same speed as the vehicle in its up-and-down motion, which is why it lands in your hand.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 31 Newton’s 1st Law So here it is, Newton’s 1st Law of Motion: An object in equilibrium (static or dynamic) will remain in equilibrium unless the net force on the object differs from zero. You may have heard other similar version, like “An object at rest remains at rest unless acted upon by an outside force”, but remember this holds for dynamic equilibrium too!  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 32 Free Body Diagrams When examining forces, it is convenient to make free body diagrams (FBDs), which is a diagram for a given mass that has all forces acting on it labelled with the correct directions. While a picture of the object is nice, forces do not care about the shape or size of the object, so a dot or box will suffice.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 33 FBDs a) The tension in the rope is 300 N, equal to Nellie’s weight. b) The tension in each rope is now 150 N, half of Nellie’s weight. In each case,Σ끫롲⃗ = 0. 끫롲  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 34 Vectors Remember that if the forces are not pointing in the same direction, we need to add them as vectors, which might mean using components! To add graphically, remember the parallelogram rule:  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 35 FBDs and Vectors Nellie’s weight is shown by the downward vertical vector. An equal and opposite vector is needed for equilibrium, shown by the dashed vector. Note that the dashed vector is the diagonal of the parallelogram defined by the dotted lines. Using the parallelogram rule, we find that the tension in each rope is more than half her weight.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 36 FBDs and Vectors As the angle between the ropes increases, tension increases so that the resultant (dashed-line vector) remains at 300 N upward, which is required to support 300-N Nellie.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 37 FBDs and Vectors When the ropes supporting Nellie are at different angles to the vertical, the tensions in the two ropes are unequal. By the parallelogram rule, we see that the right rope bears most of the load and has the greater tension.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 38 FBDs and Vectors You can safely hang from a clothesline hanging vertically, but you will break the clothesline if it is strung horizontally. This is also how you can get your car out of the mud!  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 39 Quick Question Two sets of swings are shown at right. If the children on the swings are of equal weights, the ropes of which swing are more likely to break?  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 40 Quick Question Two sets of swings are shown at right. If the children on the swings are of equal weights, the ropes of which swing are more likely to break? Girl on right  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 41 Inertia Newton’s 1st Law (NL1) is also called the “Law of Inertia”. Inertia is a tendency for an object to keep doing what it’s doing. The quantity we use to measure inertia is the object’s mass (measured in kilograms). The more mass an object has, the harder it is to change its motion  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 42 Mass Mass: The quantity of matter in an object. It is also the measure of the inertia Inertia is resistance to change in motion Greater Inertia = Greater Mass The pillow has a larger size (volume) but a smaller mass than the battery. Note that volume is the amount of space an object takes up, not how much mass it has.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 43 Mass vs. Weight Also, be sure not to confuse mass with weight. Weight is the gravitational force on an object (in deep space, your weight would be zero) Standard unit of force is the newton (N) Mass is the amount of stuff inside an object, and doesn’t depend on the local gravitational field. Unit of measurement is the kilogram (kg)  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 44 Mass vs. Weight One way to measure mass is to shake an object back and forth (this is how astronauts “weigh” themselves in space). It’s just as difficult to shake a stone in its weightless state in space as it is in its weighted state on Earth.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 45 Mass vs. Weight One kilogram of mass weighs 10 Newtons on Earth. If you prefer, a kilogram weighs 2.2 pounds on Earth The Imperial unit for mass is the stone, which weighs 14 pounds (I told you these units were stupid!) If we want to calculate the gravitational force at other locations, we use 끫롲 끫뢨끫뢨 = 끫뢴 끫뢴끫뢴끫뢴,where 끫뢴 끫뢴is the gravitational acceleration at that location. So weight and mass are proportional, but weight depends on the gravitational field strength!  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 46 Quick Question Does a 2-kilogram bunch of bananas have twice as much inertia as a 1-kilogram loaf of bread? Twice as much mass? Twice as much volume? Twice as much weight, when weighed in the same location?  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 47 Quick Question Does a 2-kilogram bunch of bananas have twice as much inertia as a 1-kilogram loaf of bread? Yes Twice as much mass? Yes Twice as much volume? No Twice as much weight, when weighed in the same location? Yes  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 48 Inertia = Mass  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 49 Quick Question When the string is pulled down slowly, the top string breaks, which best illustrates the A. weight of the ball. B. mass of the ball. C. volume of the ball. D. density of the ball.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 50 Quick Question When the string is pulled down slowly, the top string breaks, which best illustrates the A. weight of the ball. B. mass of the ball. C. volume of the ball. D. density of the ball The tension in the top string is the pulling tension in the bottom string plus the weight of the ball. So, the top string has greater tension and breaks first.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 51 Newton’s 2nd Law What if the net force is not zero (meaning the system is not in mechanical equilibrium)? Newton’s 2nd Law says ⃗끫롲 = 끫뢴끫뢜 Σ끫롲 ⃗ 끫뢜 Here 끫뢴 is the mass and 끫뢜 ⃗ is the resulting acceleration. So this is the connection between kinematics and dynamics: forces cause accelerations Acceleration is inversely proportional to mass. 1 Acceleration ∼ mass  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 52 Newton’s 2nd Law This also means the acceleration is equal to the net force divided by the mass. If the net force acting on an object doubles, its acceleration is doubled. If the mass is doubled, then acceleration will be halved. If both the net force and the mass are doubled, the acceleration will be unchanged.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 53 Newton's second law of motion Newton's second law relates acceleration, force and mass: The acceleration produced by a net force on an object is directly proportional to the net force, is in the same direction as the net force, and is inversely proportional to the mass of the object. In equation form: Acceleration = net force mass  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 54 Quick Question If a car can accelerate at 2 m/s2, what acceleration can it attain if it is towing another car of equal mass?  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 55 Quick Question If a car can accelerate at 2 m/s2, what acceleration can it attain if it is towing another car of equal mass? 1 m/s2 Explanation F = ma a = F/m = 2 m/s2 But now m = 2m a = F/2m = 2 /2 = 1 m/s2  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 56 Quick Question How much force, or thrust, must a 30,000-kg jet plane develop to achieve an acceleration of 1.5 m/s2?  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 57 Quick Question How much force, or thrust, must a 30,000-kg jet plane develop to achieve an acceleration of 1.5 m/s2? 45 kN F = ma F = (30,000)(1.5) = 45,000N F = 45 kN  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 58 Newton's second law of motion  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 59 Free Fall Now we can explain why (in the absence of air resistance) all masses fall at the same rate. Even though the force of gravity is larger for a larger mass, its mass is also larger by the same factor, so the acceleration (which is the ratio 끫롲끫롲/끫뢴끫뢴) is the same!  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 60 Quick Question A cart is pushed and undergoes a certain acceleration. Consider how the acceleration would compare if it were pushed with twice the net force while its mass increased by four. Then its acceleration would be A. one quarter. B. half. C. twice. D. the same.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 61 Quick Question A cart is pushed and undergoes a certain acceleration. Consider how the acceleration would compare if it were pushed with twice the net force while its mass increased by four. Then its acceleration would be A. one quarter. B. half. C. twice. D. the same.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 62 Quick Question A 5-kg iron ball and a 10-kg iron ball are dropped from rest. For negligible air resistance, the acceleration of the heavier ball will be A. less. B. the same. C. more. D. undetermined.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 63 Quick Question A 5-kg iron ball and a 10-kg iron ball are dropped from rest. For negligible air resistance, the acceleration of the heavier ball will be A. less. B. the same. C. more. D. undetermined. Both are in "free fall." Hence their equal acceleration  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 64 Quick Question A 5-kg iron ball and a 10-kg iron ball are dropped from rest. When the free-falling 5-kg ball reaches a speed of 10 m/s, the speed of the free-falling 10-kg ball is A. less than 10 m/s. B. 10 m/s. C. more than 10 m/s. D. undetermined.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 65 Quick Question A 5-kg iron ball and a 10-kg iron ball are dropped from rest. When the free-falling 5-kg ball reaches a speed of 10 m/s, the speed of the free-falling 10-kg ball is A. less than 10 m/s. B. 10 m/s. C. more than 10 m/s. D. undetermined. Both are in "free fall." Hence their equal speeds.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 66 Nonfree Fall When an object falls downward through the air it experiences force of gravity pulling it downward. air drag force acting upward. The condition of nonfree fall occurs when air resistance is nonnegligible. depends on two things:  speed and  frontal surface area.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 67 Nonfree fall When the object is moving fast enough so that air resistance builds up to balance the force of gravity. Then no net force No acceleration Velocity does not change Terminal velocity when air resistance balances weight, net force is zero occurs when acceleration terminates refers to terminal speed along with the direction of motion, which is downward.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 68 Air Resistance If there is drag, the acceleration will not be a constant 끫뢴끫뢴. Drag is proportional to the speed of an object: the faster it goes, the greater the drag force. So the acceleration of a falling object would start at 끫뢴 끫뢴 , but would drop off to zero as the object speeds up. When the acceleration finally reaches zero (so that the weight and the drag force have equal magnitude), the velocity the object has is called the terminal velocity.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 69 Skydiver in fall after jumping from a plane. Weight and air resistance act on the falling object. As falling speed increases, air resistance on diver builds up, net force is reduced, and acceleration is reduced. When air resistance equals the diver's weight, net force is zero and acceleration terminates. Diver reaches terminal velocity, then continues the fall at constant speed.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 70 Quick Question When a 20-N falling object encounters 5 N of air resistance, its acceleration of fall is A. less than g. B. more than g. C. g. D. terminated.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 71 Quick Question When a 20-N falling object encounters 5 N of air resistance, its acceleration of fall is A. less than g. B. more than g. C. g. D. terminated. Acceleration of a nonfree fall is always less than g. Acceleration will actually be (20N − 5 N)/2 kg = 7.5 m/s2.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 72 Quick Question Consider a heavy and a light person jumping together with same-size parachutes from the same altitude. Who will reach the ground first? A. The light person B. The heavy person C. Both will reach at the same time. D. Not enough information is provided.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 73 Quick Question Consider a heavy and a light person jumping together with same-size parachutes from the same altitude. Who will reach the ground first? A. The light person B. The heavy person C. Both will reach at the same time. D. Not enough information is provided. Both people feel a similar upward drag force at any given speed. The heavier person has a greater downward weight force than the lighter person. Thus, the heavier person must drop farther before the upward drag force balances the downward weight force, resulting in a greater terminal speed.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 74 Quick Question If the rock and the feather are dropped in a space where the air has been removed by a vacuum pump, A. the feather hits the bottom first, before the rock hits. B. the rock hits the bottom first, before the feather hits. C. the rock and the feather drop together side by side. D. more information is needed to determine whether the rock or the feather hits bottom first.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 75 Quick Question If the rock and the feather are dropped in a space where the air has been removed by a vacuum pump, A. the feather hits the bottom first, before the rock hits. B. the rock hits the bottom first, before the feather hits. C. the rock and the feather drop together side by side. D. more information is needed to determine whether the rock or the feather hits bottom first. There is no air, because it is vacuum. Hence no air resistance.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 80 Newton’s 3rd Law Newton’s 3rd Law states that To every action there is always an opposed equal reaction. It is important to note that these action-reaction pairs of forces always act on different objects!  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 81 Newton’s 3rd Law The interaction that drives the nail is the same as the one that halts the hammer.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 82 Newton’s 3rd Law In every interaction, the forces always occur in pairs. You push against the floor, and the floor simultaneously pushes against you. The tires of a car interact with the road to produce the car’s motion. The tires push against the road, and the road simultaneously pushes back on the tires. When swimming, you push the water backward, and the water pushes you forward. When the girl jumps to shore, the boat moves backward.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 83 Action-Reaction Pairs When action is A exerts force on B, the reaction is simply B exerts force on A.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 84 Action-Reaction Pairs Earth is pulled up by the boulder with just as much force as the boulder is pulled down by Earth. So why doesn’t Earth “jump up” to meet the boulder?  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 85 Action-Reaction Pairs Just because the forces are the same in magnitude, the accelerations are certainly not! Although the pair of forces between the boulder and Earth is the same, the masses are quite unequal. Acceleration is not only proportional to the net force, but it is also inversely proportional to the mass. Because Earth has a huge mass, we don’t sense its infinitesimally small acceleration.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 86 Action-Reaction Pairs The cannonball undergoes more acceleration than the cannon because its mass is much smaller.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 87 Action-Reaction Pairs F represents both the action and reaction forces; m (large), the mass of the cannon; and m (small), the mass of the cannonball. Do you see why the change in the velocity of the cannonball is greater compared with the change in velocity of the cannon?  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 88 Action-Reaction Pairs The balloon recoils from the escaping air and climbs upward. If a balloon is released and allowed to move, it accelerates as the air comes out. A rocket accelerates in much the same way—it continually recoils from the exhaust gases ejected from its engine. Each molecule of exhaust gas acts like a tiny molecular cannonball shot downward from the rocket.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 89 Quick Question A tug of war occurs between boys and girls on a polished floor that’s somewhat slippery. If the boys are wearing socks and the girls are wearing rubber-soled shoes, who will surely win, and why?  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 90 Quick Question A tug of war occurs between boys and girls on a polished floor that’s somewhat slippery. If the boys are wearing socks and the girls are wearing rubber-soled shoes, who will surely win, and why? Girls As no friction force on boys means net force is not zero  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 91 Action-Reaction Pairs So why don’t these pairs just cancel each other out and ⃗끫롲 = 0? produce Σ끫롲 We need to be careful about the system we are examining.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 92 Defining Systems A force acts on the orange, and the orange accelerates to the right. The dashed line surrounding the orange encloses and defines the system.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 93 Defining Systems The vector that pokes outside the dashed line represents an external force on the system. The system (that is, the orange) accelerates in accord with Newton’s second law. The force on the orange, provided by the apple, is not cancelled by the reaction force on the apple. The orange still accelerates.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 94 Defining Systems The force is provided by an apple, which doesn’t change our analysis. The apple is outside the system. The fact that the orange simultaneously exerts a force on the apple, which is external to the system, may affect the apple (another system), but not the orange. You can’t cancel a force on the orange with a force on the apple. So in this case the action and reaction forces don’t cancel.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 95 Defining Systems a) Action and reaction forces cancel. b) When the floor pushes on the apple (reaction to the apple’s push on the floor), the orange-apple system accelerates.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 96 Defining Systems When the force pair is internal to the orange-apple system, the forces do cancel each other. They play no role in accelerating the system. A force external to the system is needed for acceleration. When the apple pushes against the floor, the floor simultaneously pushes on the apple—an external force on the system. The system accelerates to the right. Inside a baseball, trillions of interatomic forces hold the ball together but play no role in accelerating the ball. They are part of action-reaction pairs within the ball, but they combine to zero. If the action-reaction forces are internal to the system, then they cancel and the system does not accelerate. A force external to the ball, such as a swinging bat provides, is needed to accelerate the ball.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 97 Horse-Cart Problem All the pairs of forces that act on the horse and cart are shown. The acceleration of the horse-cart system is due to the net force F – f.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 98 Horse-Cart Problem Will the horse’s pull on the cart be canceled by the opposite and equal pull by the cart on the horse, thus making acceleration impossible? From the farmer’s point of view, the only concern is with the force that is exerted on the cart system. The net force on the cart, divided by the mass of the cart, is the acceleration. The farmer doesn’t care about the reaction on the horse.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 99 Horse-Cart Problem Now look at the horse system. The opposite reaction force by the cart on the horse restrains the horse. Without this force, the horse could freely gallop to the market. The horse moves forward by interacting with the ground. When the horse pushes backward on the ground, the ground simultaneously pushes forward on the horse.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 100 Horse-Cart Problem Look at the horse-cart system as a whole. The pull of the horse on the cart and the reaction of the cart on the horse are internal forces within the system. They contribute nothing to the acceleration of the horse-cart system. They cancel and can be neglected. To move across the ground, there must be an interaction between the horse-cart system and the ground. It is the outside reaction by the ground that pushes the system.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 101 Quick Question What is the net force that acts on the cart? On the horse? On the ground?  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 102 Quick Question What is the net force that acts on the cart? ∑ 끫룚 = 끫룮 − 끫뤎 On the horse? ∑ 끫룚끫룞 = 끫룮 − 끫룚 On the ground? ∑ 끫룚끫룜 = 끫뤎 − 끫룚  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 103 Action-Reaction Pairs If a sheet of paper is held in midair, the heavyweight champion of the world could not strike the paper with a force of 200 N (45 pounds). The paper is not capable of exerting a reaction force of 200 N, and you cannot have an action force without a reaction force. If the paper is against the wall, then the wall will easily assist the paper in providing 200 N of reaction force, and more if needed!  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 104 Vectors Vector components Vertical and horizontal components of a vector are perpendicular to each other. Determined by resolution. As the stone is released, its vertical component and its horizontal component are given as vertical and horizontal vectors. The velocity of the stone continues from the release point, going between the other components. A dashed line rises and falls below the velocity component of the stone.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 105 Quick Question Nellie Newton pulls on the sled as shown. Which component of her force F is greater? What two other forces (not shown) act on the sled?  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 106 Quick Question Nellie Newton pulls on the sled as shown. Which component of her force F is greater? The horizontal component Fx is greater. What two other forces (not shown) act on the sled? Weight mg and normal N also act on the sled.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 107 Summary So, our study of forces so far amounts to knowing Newton’s Three Laws: 1. (Law of Inertia) An object will remain in equilibrium unless the net force on that object is not zero. 2. If the net force is not zero, then it will accelerate according to Σ끫롲⃗끫롲 = 끫뢴끫뢜 ⃗. 끫뢜 3. For every (action) force, there is a (reaction) force of equal magnitude but opposite direction.  Physics 110 - Lecture 2 - Summer 2024 - Sonia Katdare 108 Summary To solve any force problem, you can just follow these steps: 1. Draw a FBD for each system (mass) of interest. Clearly label all forces as vectors with their directions. 2. Add up the forces (possibly with components in each ⃗끫롲. direction) to get Σ끫롲 3. Use Newton’s 2nd Law Σ끫롲 ⃗끫롲 = 끫뢴끫뢜 (or, if in equilibrium, Newton’s 1st Law with 끫뢜 ⃗ = 0).  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 1 ROTATION, GRAVITY AND PROJECTILE MOTION Chapter 8,9,10  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 2 ROTATION Chapter 8  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 3 Student learning objectives Circular Motion Centripetal and Centrifugal Force Torque Center of Mass and Center of Gravity Rotational Inertia  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 4 Rotation and Revolution An axis is the straight line around which rotation takes place. When an object turns about an internal axis, the motion is called rotation, or spin. When an object turns about an external axis, the motion is called revolution.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 5 Rotational Speed The turntable rotates around its axis while a ladybug sitting at its edge revolves around the same axis. Which part of the turntable moves faster—the outer part where the ladybug sits or a part near the orange center? It depends on whether you are talking about linear speed or rotational speed.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 6 Types of Speed Linear speed is the distance traveled per unit of time. A point on the outer edge of the turntable travels a greater distance in one rotation than a point near the center. The linear speed is greater on the outer edge of a rotating object than it is closer to the axis. The speed of something moving along a circular path can be called tangential speed because the direction of motion is always tangent to the circle. Rotational speed (sometimes called angular speed) is the number of rotations per unit of time. All parts of the rigid turntable rotate about the axis in the same amount of time. All parts have the same rate of rotation, or the same number of rotations per unit of time. It is common to express rotational speed in revolutions per minute (RPM).  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 7 Rotational Speed All parts of the turntable rotate at the same rotational speed. a) A point farther away from the center travels a longer path in the same time and therefore has a greater tangential speed. b) A ladybug sitting twice as far from the center moves twice as fast.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 8 Rotational Speed We may calculate the tangential speed via 끫룆 = 끫뢾끫뢾 Here 끫뢾 is how far out from the axis we are, and 끫뢾 (this is the Greek letter omega, NOT “w”!) is the angular speed. You move faster if the rate of rotation increases (bigger 끫뢾). You also move faster if you are farther from the axis (bigger 끫뢾).  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 9 Quick Question At an amusement park, you and a friend sit on a large rotating disk. You sit at the edge and have a rotational speed of 4 RPM and a linear speed of 6 m/s. Your friend sits halfway to the center. a) What is her rotational speed? b) What is her linear speed?  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 10 Quick Question At an amusement park, you and a friend sit on a large rotating disk. You sit at the edge and have a rotational speed of 4 RPM and a linear speed of 6 m/s. Your friend sits halfway to the center. a) What is her rotational speed? 4 RPM b) What is her linear speed? 3 m/s  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 11 Centripetal Force We already know that an object moving in a circle is accelerating, since the velocity is changing direction. This type of acceleration, which makes an object move in a circle, is called centripetal acceleration 끫뢜⃗끫뢠. Centripetal means “center seeking”, so 끫뢜⃗끫뢠 points towards the center of the circle. Whatever force is causing this motion is called a centripetal force.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 12 Centripetal Force If you whirl a can on a string, the tension in the string is a centripetal force acting toward your hand.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 13 Centripetal Force Centripetal force holds a car in a curved path. a) For the car to go around a curve, there must be sufficient friction to provide the required centripetal force. b) If the force of friction is not great enough, skidding occurs.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 14 Centripetal Force The clothes in a washing machine are forced into a circular path, but the water is not, and it flies off tangentially.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 15 Centripetal Force Centripetal Force depends upon mass of object. tangential speed of the object. radius of the circle. Centripetal acceleration, depends on how fast the object is moving (linear speed) and the radius of the circle: 끫룆 2 끫뢜끫뢠 = = 끫뢾2 끫뢾 끫뢾 A faster turn or a tighter circle needs greater acceleration. 끫룆 2 끫롲끫뢠 = 끫뢴 = 끫뢴끫뢾2 끫뢾 끫뢾  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 16 Conical Pendulum For an extended example, let’s look at a conical pendulum (a bob that revolves in a circle, whose string sweeps out a cone). There are two forces on the bob: The tension 끫뢎 in the string The weight 끫롲끫뢨 = 끫뢴끫뢴  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 17 Conical Pendulum The tension can be resolved (broken into) its horizontal and vertical components, 끫뢎 = (끫뢎끫룊 , 끫뢎끫료 )  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 18 Conical Pendulum We drew our FBD, so now let’s find the net force: Σ끫롲끫룊 = 끫뢎끫룊 Σ끫롲끫료 = 끫뢎끫료 − 끫뢴끫뢴 We know that the acceleration in the 끫료 direction is zero since the bob is not moving up or down, just in a horizontal circle. And since it’s moving in a circle, the horizontal acceleration is centripetal: 끫뢴끫룆 2 끫뢎끫룊 = 끫뢴끫뢜끫뢠 = 끫뢾 끫뢎끫료 − 끫뢴끫뢴 = 0 Remember that 끫뢾 is the radius of the circle, not the length of the string!  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 19 Banked Curve For a car rounding a banked curve, it’s the horizontal component of the normal force (and possibly some friction) that provides the centripetal acceleration.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 20 “Centrifugal” Force Sometimes people talk about a “centrifugal” force that pushes objects away from the center of circular motion. This is not a real force! This is just inertia at work: When the string breaks (so that the centripetal force of tension goes away), the can moves in a straight line tangent to where the string broke. It does not “bend away” from the center through some “centrifugal” force.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 21 Quick Question A heavy iron ball is attached by a spring to a rotating platform, as shown in the sketch. Two observers, one in the rotating frame and one on the ground at rest, observe its motion. Which observer sees the ball being “pulled” outward, stretching the spring? Which observer sees the spring pulling the ball into circular motion?  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 22 Quick Question A heavy iron ball is attached by a spring to a rotating platform, as shown in the sketch. Two observers, one in the rotating frame and one on the ground at rest, observe its motion. Which observer sees the ball being “pulled” outward, stretching the spring? Which observer sees the spring pulling the ball into circular motion? Observer in rotating frame; observer in rest frame  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 23 Quick Question When you roll a tapered cup across a table, the path of the cup curves because the wider end rolls A. slower. B. at the same speed as the narrow part. C. faster. D. in an unexplained way.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 24 Quick Question When you roll a tapered cup across a table, the path of the cup curves because the wider end rolls A. slower. B. at the same speed as the narrow part. C. faster. D. in an unexplained way.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 25 “Centrifugal” Force The feeling of a “centrifugal” force only arises because the observer is in an accelerating reference frame. In an inertial reference frame (non-accelerating), such fictitious forces are absent. Another such example is the Coriolis force.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 26 Torque To make something rotate, a torque must be applied. Torque is produced by a turning force applied over a lever arm Forces cause acceleration; torques cause rotational acceleration  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 27 Torque Only the perpendicular component of the applied force contributes to the torque The greater the lever arm, the greater the torque about the hinge  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 28 Torque Mathematically, the torque 끫븞 (Greek letter “tau”) can be found via 끫븞 = 끫롲⊥ 끫뢾 Here 끫롲⊥ is the perpendicular component of the force, and 끫뢾 is the lever arm (“leverage”)  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 29 Quick Question If you cannot exert enough torque to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench handle as shown?  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 30 Quick Question If you cannot exert enough torque to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench handle as shown? No  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 31 Torque Torque is a vector, but kind of a strange one To find the direction, use the Right Hand Rule:  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 32 Torque This means that torque vectors usually point either out of the page (positive) or into the page (negative) You can also tell the sign based on which way it wants to rotate the object about the hinge/pivot: CCW is a positive torque CW is a negative torque To find the total rotational acceleration, we’ll need the net torque, just like with forces!  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 33 Balanced Torques A pair of torques can balance each other. Balance is achieved if the torque that tends to produce clockwise rotation by the boy equals the torque that tends to produce counterclockwise rotation by the girl: Σ끫븞⃗ = 0.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 34 Quick Question What is the weight of the block hung at the 10-cm mark?  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 35 Quick Question What is the weight of the block hung at the 10-cm mark? 15N  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 36 Quick Question Suppose the girl on the left, who weighs 200 N, is handed a 40 N bag of apples. Where on the plank should her seat be hung to maintain balance, assuming the boy does not move? A. 1 m from pivot B. 1.5 m from pivot C. 2 m from pivot D. 2.5 m from pivot  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 37 Quick Question Suppose the girl on the left, who weighs 200 N, is handed a 40 N bag of apples. Where on the plank should her seat be hung to maintain balance, assuming the boy does not move? A. 1 m from pivot B. 1.5 m from pivot C. 2 m from pivot D. 2.5 m from pivot ( lever arm )= ×F ( lever arm ) new new old × Fold ( lever arm ) new × ( 240 N) = ( 3 m ) × ( 200 N) 5 ( lever arm ) new = × ( 3 m) = 2.5 m 6  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 38 Rotational Equilibrium We will find that each of Newton’s Laws has a rotational analogue. The 1st Law is about rotational equilibrium (Σ끫븞⃗ = 0) An object in (static or dynamic) rotational equilibrium will remain in that state until the net torque differs from zero.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 39 Center of Mass The center of mass of an object is the object’s average position of mass We can pretend that all of the object’s mass is located at this one point  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 40 Center of Mass Objects of variable sizes and densities may have centers of mass that are different from their geometric centers  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 41 Center of Gravity The center of gravity of an object is where the net gravitational force acts on it. In nearly all circumstances, it’s the same location as the center of mass We can find its location with a plumb bob.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 42 Center of Gravity Sometimes there may not actually be material at the CG!  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 43 CG and Torque The block topples when the CG extends beyond its support base.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 44 CG and Torque The Leaning Tower of Pisa does not topple because its CG does not extend beyond its base. A vertical line below the CG falls inside the base, and so the Leaning Tower has stood for centuries. If the tower leaned far enough that the CG extended beyond the base, an unbalanced torque would topple the tower.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 46 CG and Torque You can lean over and touch your toes without toppling only if your CG is above the area bounded by your feet.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 47 Stability a) Equilibrium is unstable when the CG is lowered with displacement. b) Equilibrium is stable when the CG is raised with displacement. c) Equilibrium is neutral when displacement neither raises nor lowers the CG.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 49 Stability The CG of an object has a tendency to take the lowest position available. a) A table tennis ball is placed at the bottom of a container of dried beans. b) When the container is shaken from side to side, the ball is nudged to the top.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 50 Axes of Rotation In general, an object can rotate about any axis, but there are three principle axes at right angles to each other  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 51 Rotational Inertia So, if mass measures inertia (how hard it is to accelerate an object), what measures rotational inertia (how hard it is to rotationally accelerate an object)? This is called the moment of inertia 끫롸 (there are actually different moments for the different axes). Moment of inertia depends not only on mass, but how far the mass is distributed from the axis of rotation.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 52 Moment of Inertia An ice skater rotates around her longitudinal axis when going into a spin. a) The skater has the least amount of rotational inertia when her arms are tucked in. b) The rotational inertia when both arms are extended is about three times more than in the tucked position. c) With your leg and arms extended, you can vary your spin rate by as much as six times.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 53 Moment of Inertia A flip involves rotation about the transverse axis. a) Rotational inertia is least in the tuck position. b) Rotational inertia is 1.5 times greater. c) Rotational inertia is 3 times greater. d) Rotational inertia is 5 times greater than in the tuck position.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 54 Moment of Inertia A solid cylinder rolls down an incline faster than a hollow one, whether or not they have the same mass or diameter. This is because the moment of the hollow cylinder is larger.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 55 Quick Question A heavy iron cylinder and a light wooden cylinder, similar in shape, roll down an incline. Which will have more acceleration?  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 56 Quick Question A heavy iron cylinder and a light wooden cylinder, similar in shape, roll down an incline. Which will have more acceleration? The same  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 57 Moment of Inertia Rotational inertias of various objects are different. (It is not important for you to learn these values, but you can see how they vary with the shape and axis.)  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 58 Quick Question Would you expect the rotational inertia of a hollow sphere about its center to be greater or less than the rotational inertia of an equal mass and radius solid sphere?  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 59 Quick Question Would you expect the rotational inertia of a hollow sphere about its center to be greater or less than the rotational inertia of an equal mass and radius solid sphere? Greater  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 60 Newton’s 2nd Law for Rotation Now that we know that torque is like rotational force and moment is like rotational mass, we can think about rotational acceleration as it relates to linear acceleration: 끫뢜 = 끫뢾끫뢾 Here 끫뢾 (Greek letter “alpha”) is the angular acceleration, or the rate of change of the angular velocity 끫뢾 The vector directions of these are found with the Right Hand Rule (RHR), same as torques  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 61 Newton’s 2nd Law for Rotation So if the linear version of NL2 is Σ끫롲⃗ = 끫뢴끫뢜⃗, the rotational version should be Σ끫븞⃗ = 끫롸끫뢾⃗ And it is! A net torque on an object will cause it to rotationally accelerate, and objects with higher moments (more mass spread out) will accelerate less.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 62 Newton’s 3rd Law for Rotation And just as with the third law for forces, Newton’s 3rd Law for rotation states For every torque object A exerts on object B, object B exerts an equal magnitude and opposite direction torque back on A. Torques, like forces, are interactions involving two objects.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 63 GRAVITY Chapter 9  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 64 Student learning objectives The Universal Law of Gravity The Universal Gravitational Constant Gravity and Distance: Inverse-Square Law Weight and Weightlessness Ocean Tides Gravitational Fields Einstein's Theory of Gravitation  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 65 Newton Isaac Newton realized that both an apple from a tree and the moon “fall” towards Earth because a force is exerted between them. In the case of the moon, the moon is also moving tangentially quickly enough that it doesn’t hit Earth  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 66 Falling If the force that pulls apples off trees also pulls the moon into orbit, the circle of the moon’s orbit should fall 1.4 mm below a point along the straight line where the moon would otherwise be one second later.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 67 Falling So moons “fall” around planets, and planets “fall” around the sun. The proper term is orbit. We can see how a cannon firing cannonballs with enough speed could put one into orbit.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 68 Law of Gravitation Law of universal gravitation: Everything pulls on everything else. Every body attracts every other body with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance separating them. 끫롴끫뢴1 끫뢴2 끫롲끫뢨 = 끫뢢2 Here 끫롴 = 6.67 × 10−11 N·m2/kg2 is Newton’s Constant, and tells us that Gravity is relatively weak (need big masses to matter) Gravity is universal (all masses attract with the same intrinsic strength)  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 69 Law of Gravitation The gravitational force gets weaker the further out you go, as an inverse square law Many other forces do the same thing! The graph plots gravitational force versus distance. At the vertex is Earth. An apple weighs 1 Newton here, and the gravitational force is 1 over d squared. At a distance of 2 d, an apple weighs one-fourth Newton. At 4 d, the apple weights blank Newtons. On the graph, a curve falls with decreasing steepness from the y-axis, approaching the x-axis.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 70 Law of Gravitation Your weight is less at the top of a mountain because you are farther from the center of Earth. Inverse-square force law: Applies to a force magnitude which varies as the inverse- square of the distance from the source of the force. At greater distances, there is a smaller force. Even at great distances, force approaches but never reaches zero.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 71 Quick Question If two masses are doubled and the distance between them tripled, what would the new gravitational force between them be?  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 72 Quick Question If two masses are doubled and the distance between them tripled, what would the new gravitational force between them be? 4/9 If both masses double. Then, 2x2=4 If distance is tripled. Then, (1/3) x (1/3) = 1/9  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 73 Force of Gravity at Surface of Earth Given the mass of Earth 끫뢀 = 6 × 1024 kg, and the radius of Earth 끫뢊 = 6.4 × 106 m, we can fill in the gravitational force for something sitting right on Earth’s surface: 끫롴끫뢀끫뢴 6.67 × 10−11 6 × 1024 끫뢴 2 끫뢴 끫롲끫뢨 = = = 9.8 m/s 끫뢊2 6.4 × 106 2 This is just 끫롲끫뢨 = 끫뢴끫뢴! So we can see that the definition of the gravitational field strength at the surface of Earth is 끫롴끫롴 끫뢴 = 2 끫뢊  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 74 Gravitational Fields Gravitational field is an alteration of space around Earth (or any object with mass). vector field is a set of vectors at every point in space. When a mass is placed into space, it creates a gravitational field everywhere around it Other masses then interact with the field where they sit  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 75 Gravitational Fields Thanks to Einstein, we have another interpretation of gravitational fields: the bending of space (and time) in the presence of matter. The warped space-time affects the motion of other objects. Analogy: a marble rolling on the waterbed “gravitates” toward the dent  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 76 Ocean Tides The differences between ocean levels at different times of the day are called tides. There are typically two high tides and two low tides each day. Ocean tides are caused by the gravitational attraction of the Moon and the Sun due to change in distance  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 77 Tides Unequal tugs on Earth's oceans causes a stretching effect that produces a pair of ocean bulges. The spinning of the Earth beneath these “tidal bulges” creates two high and two low tides per day. Note that it’s the difference in forces that produce the tides; the sun has a greater force on Earth, but less effect on the tides  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 78 PROJECTILE AND SATELLITE MOTION Chapter 10  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 79 Student learning objectives Projectile Motion Fast-Moving Projectiles – Satellites Circular Satellite Orbits Elliptical Orbits Satellite Motion Escape Speed  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 80 Projectile Motion In 1D, object has either vertical (with gravity) or horizontal motion (without gravity) Projectile is any object tossed at an angle that follows a curved path. It moves through space under the influence of gravity, continuing in motion by its own inertia  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 81 Projectile Motion The horizontal direction has no acceleration, and the vertical direction has acceleration 끫뢴 Here a dropped ball and a horizontally launched ball fall the same amount in time.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 82 Projectile Motion No matter the angle at which a projectile is launched, the vertical distance of fall beneath the dashed straight-line path is the same for equal times.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 83 Projectile Motion With no gravity the projectile would follow the straight-line path (dashed line). But because of gravity it falls beneath this line the same vertical distance it would fall if it were released from rest.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 84 Projectile Motion The velocity of a projectile is shown at various points along its path. Vertical component changes while the horizontal component does not. Velocity at any point is computed with the Pythagorean theorem (diagonal of rectangle).  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 85 Projectile Motion The paths of projectiles launched at the same speed but at different angles. The paths neglect air resistance.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 86 Projectile Motion In the presence of air resistance, the path of a high-speed projectile falls below the idealized parabola and follows the solid curve.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 87 Projectile Motion Without air resistance, the time for a projectile to reach maximum height is the same as the time for it to return to its initial level.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 88 Quick Question A projectile is launched at an angle into the air. Neglecting air resistance, what is its vertical acceleration? Its horizontal acceleration?  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 89 Quick Question A projectile is launched at an angle into the air. Neglecting air resistance, what is its vertical acceleration? -g Its horizontal acceleration? 0  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 90 Fast-Moving Projectiles—Satellites Satellite motion is an example of a high-speed projectile. A satellite is simply a projectile that falls around Earth rather than onto it. Sufficient tangential velocity needed for orbit. With no air resistance to reduce speed, a satellite orbits Earth indefinitely.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 91 Satellite Motion How fast? Well, the only force on the object is the gravitational force, and we need a circular orbit right at the radius of Earth. That means the gravitational force should provide a centripetal acceleration: 끫롴끫뢀끫뢴 끫뢴끫룆 2 끫롲끫뢨 = 2 = 끫뢴끫뢜끫뢠 = 끫뢊 끫뢊 We can do some algebra to solve for 끫룆: 끫롴끫뢀 끫룆 = = 8 km/s 끫뢊  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 92 Satellite Motion A real object orbiting at a height of zero meters above the Earth’s surface would hit mountains and have incredible drag force. Actual satellites need to orbit at a height of 150 km to avoid the atmosphere. We can also calculate the period of an orbit (time to complete one revolution) using distance over speed: 2끫븖끫뢊 끫뢎 = 끫룆 The higher the altitude, the greater the period. At about 6.5 Earth radii, a satellite would have a period of 24 hours; these are called geosynchronous satellites.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 93 Satellite Motion Circular Satellite moves in a direction perpendicular to the force of gravity acting on it Period for complete orbit about Earth about 90 minutes for ISS about 12 hours for GPS satellites about 27 days for the Moon  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 94 Satellite Motion In general, orbits are in the shape of ellipses, with one focus at the center of the planet. The parabolic paths of projectiles, such as cannonballs, are actually segments of ellipses. a) For relatively low speeds, the center of Earth is the far focus. b) For greater speeds, the near focus is Earth’s center.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 95 Quick Question A girl throws a rock horizontally from a cliff. One second after she releases the rock it will have fallen vertically by A. 1 meter. B. 5 meters. C. 10 meters. D. None of the above.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 96 Quick Question A girl throws a rock horizontally from a cliff. One second after she releases the rock it will have fallen vertically by A. 1 meter. Regardless of the initial horizontal speed, B. 5 meters. the ball will fall a vertical distance of 5 meters below the dashed line in the first C. 10 meters. second. D. None of the above.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 97 Quick Question The orbit of a satellite is shown in the sketch. In which of the positions A through D does the satellite have the greatest speed? The least speed?  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 98 Quick Question The orbit of a satellite is shown in the sketch. In which of the positions A through D does the satellite have The greatest speed? A The least speed? C  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 99 Satellite Motion If an object has enough speed, it can actually leave the planet and never come back; it switched from an elliptical orbit to a parabolic or hyperbolic one The escape speed is the needed speed to do this 2끫롴끫뢀 끫룆끫뢤 = 끫뢊  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 100 Escape Velocity For Earth, this is about 11 km/s For the Sun (solar system) at Earth’s distance, this is about 42 km/s. Several space probes have left the solar system with these speeds, and now travel through interstellar space  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 101 Escape Velocity When a projectile achieves escape speed from Earth, it A. forever leaves Earth's gravitational field. B. outruns the influence of Earth's gravity, but is never beyond it. C. comes to an eventual stop, returning to Earth at some future time. D. reaches a maximal height and then enters a perpetual orbit.  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 102 Escape Velocity When a projectile achieves escape speed from Earth, it A. forever leaves Earth's gravitational field. B. outruns the influence of Earth's gravity, but is never beyond it. C. comes to an eventual stop, returning to Earth at some future time. D. reaches a maximal height and then enters a perpetual orbit.  -Physics 110 - Lecture 4 - Sonia Katdare 1 MOMENTUM AND ENERGY Chapter 6, 7  Physics 110 - Lecture 4 - - Sonia Katdare 2 MOMENTUM Chapter 6  Physics 110 - Lecture 4 - - Sonia Katdare 3 Student Learning Objectives Momentum Impulse Impulse Changes Momentum Bouncing Conservation of Momentum Collisions More Complicated Collisions  Physics 110 - Lecture 4 - - Sonia Katdare 4 Momentum While mass is a great measure of inertia, there is something extra that’s important for moving objects… A fast moving ball is harder to stop than a slowly moving ball, so speed must be important. Let’s define the momentum of an object as its mass times its velocity: ⃗ = 끫뢴 끫룆 끫뢺 ⃗ We use the letter 끫뢺 because we’ve run out of m’s!  Physics 110 - Lecture 4 - - Sonia Katdare 5 Momentum A truck rolling down a hill has more momentum than a roller skate with the same speed. But if the truck is at rest and the roller skate moves, then the skate has more momentum.  Physics 110 - Lecture 4 - - Sonia Katdare 6 Quick Question When the speed of an object is doubled, its momentum A. remains unchanged in accord with the conservation of momentum. B. doubles. C. quadruples. D. decreases.  Physics 110 - Lecture 4 - - Sonia Katdare 7 Quick Question When the speed of an object is doubled, its momentum A. remains unchanged in accord with the conservation of momentum. B. doubles. C. quadruples. D. decreases.  Physics 110 - Lecture 4 - - Sonia Katdare 8 Impulse If the momentum of an object changes, either the mass, the velocity, or both must change. We call the change in momentum the impulse delivered to the object: ⃗끫롺 = Δ끫뢺 ⃗ 끫뢺 To change the momentum of an object, a force must act on it for some time, so we also have ⃗끫롺 = ⃗끫롲Δ끫룂  Physics 110 - Lecture 4 - - Sonia Katdare 9 Impulse When you push with the same force for twice the time, you impart twice the impulse and produce twice the change in momentum.  Physics 110 - Lecture 4 - - Sonia Katdare 10 Impulse To increase the momentum of an object, apply the greatest force possible for as long as possible. A golfer teeing off and a baseball player trying for a home run do both of these things when they swing as hard as possible and follow through with their swing. ⃗ Δ끫룂 = Δ끫뢺 Σ끫롲 끫롲 ⃗ = Δ(끫뢴끫룆 끫뢺 ⃗) 끫룆  Physics 110 - Lecture 4 - - Sonia Katdare 11 Impulse The force of impact on a golf ball varies throughout the duration of impact. A golf club that strikes a golf ball exerts zero force on the ball until it comes in contact with it. The force increases rapidly as the ball becomes distorted. The force diminishes as the ball comes up to speed and returns to its original shape. We can use the average force to solve for the impulse on an object.  Physics 110 - Lecture 4 - - Sonia Katdare 12 Impulse If the change in momentum occurs over a long time, the force of impact is small. If the change in momentum occurs over a short time, the force of impact is large.  Physics 110 - Lecture 4 - - Sonia Katdare 13 Impulse When you extend the time, you reduce the force. A padded dashboard in a car is safer than a rigid metal one. Airbags save lives. To catch a fast-moving ball, extend your hand forward and move it backward after making contact with the ball. When you jump down to the ground, bend your knees when your feet make contact with the ground to extend the time during which your momentum decreases. A wrestler thrown to the floor extends his time of hitting the mat, spreading the impulse into a series of smaller ones as his foot, knee, hip, ribs, and shoulder successively hit the mat.  Physics 110 - Lecture 4 - - Sonia Katdare 14 Quick Question When a dish falls, will the impulse be less if it lands on a carpet than if it lands on a hard floor?  Physics 110 - Lecture 4 - - Sonia Katdare 15 Quick Question When a dish falls, will the impulse be less if it lands on a carpet than if it lands on a hard floor? No, but the force will different The same initial momentum is reduced to zero in both cases. Thus, the change in momentum, and therefore the impulse, is the same in both cases. The average force is less with the carpet than with the floor because soft carpet allows a longer time for the dish to come to rest.  Physics 110 - Lecture 4 - - Sonia Katdare 16 Quick Question When a car is out of control, would it be better to hit a haystack or concrete wall?  Physics 110 - Lecture 4 - - Sonia Katdare 17 Quick Question When a car is out of control, it is better to hit a haystack than a concrete wall. Same impulse either way, but extension of hitting time reduces the force.  Physics 110 - Lecture 4 - - Sonia Katdare 18 Quick Question If a boxer is able to make the contact time five times longer by “riding” with the punch, how much will the force of the punch impact be reduced?  Physics 110 - Lecture 4 - - Sonia Katdare 19 Quick Question If a boxer is able to make the contact time five times longer by “riding” with the punch, how much will the force of the punch impact be reduced? By a factor of 5 In figure a, t is written larger than the f so punch is weak. In figure b, F is larger than the t so punch is hard.  Physics 110 - Lecture 4 - - Sonia Katdare 20 Bouncing For an object to bounce, its momentum must first come to zero, and then be accelerated back in the opposite direction This means the impulse delivered to a bouncing object is more (up to twice) than an object simply brought to rest  Physics 110 - Lecture 4 - - Sonia Katdare 21 Bouncing  Physics 110 - Lecture 4 - - Sonia Katdare 22 Bouncing The waterwheels used in gold mining operations during the California Gold Rush were not very effective. Lester A. Pelton designed a curve-shaped paddle that caused the incoming water to make a U-turn upon impact. The water “bounced,” increasing the impulse exerted on the waterwheel.  Physics 110 - Lecture 4 - - Sonia Katdare 23 Conservation of Momentum The force (impulse) that changes a system’s momentum must be an external one. If no external force is exerted on a system, the momentum doesn’t change: we say the momentum is conserved.  Physics 110 - Lecture 4 - - Sonia Katdare 24 Conservation of Momentum The force on the cannonball inside the cannon barrel is equal and opposite to the force causing the cannon to recoil. The action and reaction forces are internal to the system so they don’t change the momentum of the cannon-cannonball system. Before the firing, the momentum is zero. After the firing, the net momentum is still zero. Net momentum is neither gained nor lost.  Physics 110 - Lecture 4 - - Sonia Katdare 25 Conservation of Momentum Remember that momentum is a vector quantity, so conservation just means that the total (net) momentum is a constant: Σ끫뢺 ⃗ = Σ끫뢺 끫뢺 ⃗′ 끫뢺 To avoid too many subscripts later, I’ll use the notation “prime is after collision, no prime is before”  Physics 110 - Lecture 4 - - Sonia Katdare 26 Quick Question Newton’s second law states that if no net force is exerted on a system, no acceleration occurs. Does it follow that no change in momentum occurs?  Physics 110 - Lecture 4 - - Sonia Katdare 27 Quick Question Newton’s second law states that if no net force is exerted on a system, no acceleration occurs. Does it follow that no change in momentum occurs? Yes net momentum before collision equals net momentum after collision. in equation form: (net mv )before = (net mv )after  Physics 110 - Lecture 4 - - Sonia Katdare 28 Collisions When two (or more) objects collide in the absence of external forces (which is usually a good approximation), the total momentum before the collision is the same as the total momentum after the collision.  Physics 110 - Lecture 4 - - Sonia Katdare 29 Elastic/Inelastic Collisions If kinetic energy is conserved in addition to momentum, the collision is called elastic. These collisions are usually between hard objects like pool balls; there is no “sticking” or “squishing” If a collision does not conserve kinetic energy, the collision is inelastic, meaning some energy goes to heat or deforming the objects. (Momentum, however, is still conserved). When two objects collide together into one object, the collision is completely inelastic.  Physics 110 - Lecture 4 - - Sonia Katdare 30 Inelastic Collision In an inelastic collision between two freight cars, the momentum of the freight car on the left is shared with the freight car on the right.  Physics 110 - Lecture 4 - - Sonia Katdare 31 Quick Question Freight car A is moving toward identical freight car B that is at rest. When they collide, both freight cars couple together. Compared with the initial speed of freight car A, the speed of the coupled freight cars is A. the same. B. half. C. twice. D. None of the above.  Physics 110 - Lecture 4 - - Sonia Katdare 32 Quick Question Freight car A is moving toward identical freight car B that is at rest. When they collide, both freight cars couple together. Compared with the initial speed of freight car A, the speed of the coupled freight cars is A. the same. B. half. C. twice. D. None of the above. After the collision, the mass of the moving freight cars has doubled. Since the momentum is the product of mass and speed, the speed must be halved to conserve the value of that product.  Physics 110 - Lecture 4 - - Sonia Katdare 33 Quick Question One glider is loaded so it has three times the mass of another glider. The loaded glider is initially at rest. The unloaded glider collides with the loaded glider and the two gliders stick together. Describe the motion (speed) of the gliders after the collision.  Physics 110 - Lecture 4 - - Sonia Katdare 34 Quick Question One glider is loaded so it has three times the mass of another glider. The loaded glider is initially at rest. The unloaded glider collides with the loaded glider and the two gliders stick together. Describe the motion (speed) of the gliders after the collision. m’ = m + 3m so the gliders move at 1/4th the initial speed  Physics 110 - Lecture 4 - - Sonia Katdare 35 Quick Question Consider a 6-kg fish that swims toward and swallows a 2-kg fish that is at rest. a) If the larger fish swims at 1 m/s, what is its velocity immediately after lunch? b) What if the small fish were swimming to the left at 3 m/s?  Physics 110 - Lecture 4 - - Sonia Katdare 36 Quick Question Consider a 6-kg fish that swims toward and swallows a 2-kg fish that is at rest. a)If the larger fish swims at 1 m/s, what is its velocity immediately after lunch? ¾ m/s b)What if the small fish were swimming to the left at 3 m/s? 0 m/s  Physics 110 - Lecture 4 - - Sonia Katdare 37 Quick Question On roller blades you horizontally toss a ball away from you. The mass of the ball is one tenth your mass. Compared with the speed you give to the ball, your recoil speed will ideally be A. one tenth as much. B. the same. C. ten times as much. D. 100 times as much.  Physics 110 - Lecture 4 - - Sonia Katdare 38 Quick Question On roller blades you horizontally toss a ball away from you. The mass of the ball is one tenth your mass. Compared with the speed you give to the ball, your recoil speed will ideally be A. one tenth as much. B. the same. C. ten times as much. D. 100 times as much.  Physics 110 - Lecture 4 - - Sonia Katdare 39 Momentum Vectors Remember that momentum is a vector, so if we have a two-dimensional collision, momentum is conserved in each component separately Σ끫뢺끫뢺끫룊끫룊 = Σ끫뢺끫뢺끫룊끫룊′ Σ끫뢺끫뢺끫료끫료 = Σ끫뢺끫뢺끫료′끫료  Physics 110 - Lecture 4 - - Sonia Katdare 40 Momentum Vectors When the firecracker bursts, the vector sum of the momenta of its fragments add up to the firecracker’s momentum just before bursting.  Physics 110 - Lecture 4 - - Sonia Katdare 41 Quick Question A falling firecracker bursts into two pieces. Compared with the momentum of the firecracker when it bursts, the two pieces A. combined have the same momentum. B. each have half as much momentum. C. have more momentum. D. may or may not have more momentum.  Physics 110 - Lecture 4 - - Sonia Katdare 42 Quick Question A falling firecracker bursts into two pieces. Compared with the momentum of the firecracker when it bursts, the two pieces A. combined have the same momentum. B. each have half as much momentum. C. have more momentum. D. may or may not have more momentum.  Physics 110 - Lecture 4 - - Sonia Katdare 2 ANGULAR MOMENTUM Chapter 8  Physics 110 - Lecture 4 - - Sonia Katdare 43 Student learning Objectives Angular Momentum Conservation of Angular Momentum  Physics 110 - Lecture 4 - - Sonia Katdare 44 Angular Momentum If (linear or translational) momentum is a measure of inertia in motion, there should be a similar quantity for a rotating object. The angular momentum of a rotating object is defined as 끫롾 = 끫롸끫븨 It depends on the moment of inertia and the angular velocity  Physics 110 - Lecture 4 - - Sonia Katdare 45 Angular Momentum An object of concentrated mass m, whirling in a circular path of radius r with a speed ω has angular momentum L=Iωr This applies to a tin can swinging from a long string or a planet orbiting in a circle around the sun.  Physics 110 - Lecture 4 - - Sonia Katdare 46 Rotational version of Newton's first law: An object or system of objects will maintain its angular momentum unless acted upon by an external net torque. An external net torque is required to change the angular momentum of an object.  Physics 110 - Lecture 4 - - Sonia Katdare 47 Quick Question Suppose you are swirling a can around and suddenly decide to pull the rope in halfway; by what factor would the speed of the can change? A. Double B. Four times C. Half D. One-quarter  Physics 110 - Lecture 4 - - Sonia Katdare 48 Quick Question Suppose you are swirling a can around and suddenly decide to pull the rope in halfway; by what factor would the speed of the can change? A. Double B. Four times C. Half D. One-quarter Angular Momentum is proportional to radius of the turn. No external torque acts with inward pull, so angular momentum is conserved. Half radius means speed doubles.  Physics 110 - Lecture 4 - - Sonia Katdare 49 Angular Impulse To change the angular momentum of an object, an angular impulse must be applied. This means an external torque must act on the object for some time: ∑ 끫븞끫룂 ⃗ = Δ끫롾 = Δ 끫롸끫븨 In the absence of external torque, angular momentum is conserved!  Physics 110 - Lecture 4 - - Sonia Katdare 50 Angular Momentum It is easier to balance on a moving bicycle than on one at rest. The spinning wheels have angular momentum. When our center of gravity is not above a point of support, a slight torque is produced. When the wheels are at rest, we fall over. When the bicycle is moving, the wheels have angular momentum, and a greater torque is required to change the direction of the angular momentum.  Physics 110 - Lecture 4 - - Sonia Katdare 51 Law of conservation of angular momentum The law of conservation of angular momentum states: If no external net torque acts on a rotating system, the angular momentum of that system remains constant. Analogous to the law of conservation of linear momentum: If no external force acts on a system, the total linear momentum of that system remains constant.  Physics 110 - Lecture 4 - - Sonia Katdare 52 Conservation of Angular Momentum When the man pulls his arms and the whirling weights inward, he decreases his rotational inertia, and his rotational speed correspondingly increases.  Physics 110 - Lecture 4 - - Sonia Katdare 53 Conservation of Angular Momentum Rotational speed is controlled by variations in the body’s rotational inertia as angular momentum is conserved during a forward somersault. This is done by moving some part of the body toward or away from the axis of rotation.  Physics 110 - Lecture 4 - - Sonia Katdare 55 ENERGY Chapter 7  Physics 110 - Lecture 4 - - Sonia Katdare 56 Student Learning Objectives Energy Work Mechanical Energy: Potential and Kinetic Work-Energy Theorem Conservation of Energy Power Machines Efficiency Recycled Energy Energy for Life Sources of Energy Energy Quantifies a system’s ability to do work Occurs in a variety of forms Observed when it is being transferred or being transformed Is a conserved quantity  Physics 110 - Lecture 4 - - Sonia Katdare 57 Work Work is the product of a force exerted through some distance (in the same direction) 끫뢔 = F.d The unit of measurement for work combines a unit of force, N, with a unit of distance, m. The unit of work is the newton-meter (N m), also called the joule. One joule (J) of work is done when a force of 1 N is exerted over a distance of 1 m (lifting an apple over your head).  Physics 110 - Lecture 4 - - Sonia Katdare 58 Quick Question Suppose that you apply a 60-N horizontal force to a 32-kg package, which pushes it 4 meters across a mailroom floor. How much work do you do on the package?  Physics 110 - Lecture 4 - - Sonia Katdare 59 Quick Question Suppose that you apply a 60-N horizontal force to a 32-kg package, which pushes it 4 meters across a mailroom floor. How much work do you do on the package? 끫룼= 끫룚. d = (mg).d 끫룼= (60)(32)(10) = 240 J Quick Question You do work when pushing a cart with a constant force. If you push the cart twice as far, then the work you do is A. less than twice as much. B. twice as much. C. more than twice as much. D. zero. Quick Question You do work when pushing a cart with a constant force. If you push the cart twice as far, then the work you do is A. less than twice as much. B. twice as much. C. more than twice as much. D. zero. 끫룼= 끫룚. d = 끫룚.2d = 2W  Physics 110 - Lecture 4 - - Sonia Katdare 60 Work Twice as much work is done in lifting 2 loads 1 story high versus lifting 1 load the same vertical distance. Reason: force needed to lift twice the load is twice as much. Twice as much work is done in lifting a load 2 stories instead of 1 story. Reason: distance is twice as great.  Physics 110 - Lecture 4 - - Sonia Katdare 61 Work Note that if a force is exerted but there is no displacement, no work is done. Also, the force must have some component parallel to the displacement; only force in the direction of motion will do work! A weightlifter raising a barbell from the floor does work on the barbell. Work done against gravity.  Physics 110 - Lecture 4 - - Sonia Katdare 62 Power Power is the rate at which work is done The unit of power is the joule per second, also known as the Watt. One watt (W) of power is expended when one joule of work is done in one second. Power A worker uses more power running up the stairs than climbing the same stairs slowly. Doubling the power of an engine doubles the work done in a particular time interval. In the United States, we customarily rate engines in units of horsepower and electricity in kilowatts, either may be used. Units for Power Horsepower is a unit of measurement for the rate at which work is done. The term was adopted in the late 18th century by Scottish engineer James Watt to compare the output of steam engines with the power of draft horses. One horsepower is the amount of power used by a horse to lift a 550 pounds heavy object from a depth of 1 foot in one second. I Hp = 746 Watts  Physics 110 - Lecture 4 - - Sonia Katdare 64 Quick Question A job can be done slowly or quickly. These methods require the same amount of work, but different amounts of A. energy. B. momentum. C. power. D. impulse.  Physics 110 - Lecture 4 - - Sonia Katdare 65 Quick Question A job can be done slowly or quickly. These methods require the same amount of work, but different amounts of A. energy. B. momentum. C. power. D. impulse. Power is the rate at which work is done, so doing the job faster requires more power.  Physics 110 - Lecture 4 - - Sonia Katdare 66 Mechanical Energy Mechanical energy is due to position or to motion, or both. When work is done by an archer in drawing back a bowstring, the bent bow acquires the ability to do work on the arrow. When work is done to raise the heavy ram of a pile driver, the ram acquires the ability to do work on the object it hits when it falls. When work is done to wind a spring mechanism, the spring acquires the ability to do work on various gears to run a clock, ring a bell, or sound an alarm.  Physics 110 - Lecture 4 - - Sonia Katdare 67 Mechanical Energy Something has been acquired that enables the object to do work. It may be in the form of a compression of atoms in the material of an object; a physical separation of attracting bodies; or