Physics 110 Lecture 3 - Rotation, Gravity, Projectiles PDF

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 lear...

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 𝑣𝑣 = 𝑟𝑟𝑟𝑟 Here 𝑟𝑟 is how far out from the axis we are, and 𝜔𝜔 (this is the Greek letter omega, NOT “w”!) is the angular speed. You move faster if the rate of rotation increases (bigger 𝜔𝜔). You also move faster if you are farther from the axis (bigger 𝑟𝑟).  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 𝑎𝑎⃗𝑐𝑐. Centripetal means “center seeking”, so 𝑎𝑎⃗𝑐𝑐 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: 𝑣𝑣 2 𝑎𝑎𝑐𝑐 = = 𝜔𝜔2 𝑟𝑟 𝑟𝑟 A faster turn or a tighter circle needs greater acceleration. 𝑣𝑣 2 𝐹𝐹𝑐𝑐 = 𝑚𝑚 = 𝑚𝑚𝜔𝜔2 𝑟𝑟 𝑟𝑟  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 𝑇𝑇 in the string The weight 𝐹𝐹𝑔𝑔 = 𝑚𝑚𝑚𝑚  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 17 Conical Pendulum The tension can be resolved (broken into) its horizontal and vertical components, 𝑇𝑇 = (𝑇𝑇𝑥𝑥 , 𝑇𝑇𝑦𝑦 )  Physics 110 - Lecture 3 - Summer 2024 - Sonia Katdare 18 Conical Pendulum We drew our FBD, so now let’s find the net force: Σ𝐹𝐹𝑥𝑥 = 𝑇𝑇𝑥𝑥 Σ𝐹𝐹𝑦𝑦 = 𝑇𝑇𝑦𝑦 − 𝑚𝑚𝑚𝑚 We know that the acceleration in the 𝑦𝑦 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: 𝑚𝑚𝑣𝑣 2 𝑇𝑇𝑥𝑥 = 𝑚𝑚𝑎𝑎𝑐𝑐 = 𝑟𝑟 𝑇𝑇𝑦𝑦 − 𝑚𝑚𝑚𝑚 = 0 Remember that 𝑟𝑟 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 𝜏𝜏 (Greek letter “tau”) can be found via 𝜏𝜏 = 𝐹𝐹⊥ 𝑟𝑟 Here 𝐹𝐹⊥ is the perpendicular component of the force, and 𝑟𝑟 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: Σ𝜏𝜏⃗ = 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 (Σ𝜏𝜏⃗ = 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 𝐼𝐼 (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: 𝑎𝑎 = 𝑟𝑟𝑟𝑟 Here 𝛼𝛼 (Greek letter “alpha”) is the angular acceleration, or the rate of change of the angular velocity 𝜔𝜔 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 Σ𝐹𝐹⃗ = 𝑚𝑚𝑎𝑎⃗, the rotational version should be Σ𝜏𝜏⃗ = 𝐼𝐼𝛼𝛼⃗ 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. 𝐺𝐺𝑚𝑚1 𝑚𝑚2 𝐹𝐹𝑔𝑔 = 𝑑𝑑 2 Here 𝐺𝐺 = 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 𝑀𝑀 = 6 × 1024 kg, and the radius of Earth 𝑅𝑅 = 6.4 × 106 m, we can fill in the gravitational force for something sitting right on Earth’s surface: 𝐺𝐺𝐺𝐺𝐺𝐺 6.67 × 10−11 6 × 1024 𝑚𝑚 2 𝐹𝐹𝑔𝑔 = 2 = 6 2 = 9.8 m/s 𝑚𝑚 𝑅𝑅 6.4 × 10 This is just 𝐹𝐹𝑔𝑔 = 𝑚𝑚𝑚𝑚! So we can see that the definition of the gravitational field strength at the surface of Earth is 𝐺𝐺𝐺𝐺 𝑔𝑔 = 2 𝑅𝑅  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 𝑔𝑔 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: 𝐺𝐺𝐺𝐺𝐺𝐺 𝑚𝑚𝑣𝑣 2 𝐹𝐹𝑔𝑔 = 2 = 𝑚𝑚𝑎𝑎𝑐𝑐 = 𝑅𝑅 𝑅𝑅 We can do some algebra to solve for 𝑣𝑣: 𝐺𝐺𝐺𝐺 𝑣𝑣 = = 8 km/s 𝑅𝑅  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: 2𝜋𝜋𝜋𝜋 𝑇𝑇 = 𝑣𝑣 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 2𝐺𝐺𝐺𝐺 𝑣𝑣𝑒𝑒 = 𝑅𝑅  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.