G10 UCM Hand Out PDF

Uniform Circular Motion change can be LESS EVENLY because the A movement of an object while rotating along a SMOOTH OR EVEN. speed is constant. It is circular path with a constant or uniform speed....

Uniform Circular Motion change can be LESS EVENLY because the A movement of an object while rotating along a SMOOTH OR EVEN. speed is constant. It is circular path with a constant or uniform speed. When the object speeds always TANGENT to the up or slows down, the circle and points in the change in direction feels direction the object is more UNEVEN. moving at that moment. The blue ball moves in a The red ball moves in a Linear Velocity (v) - The rate at which an object uniform circular motion, nonuniform circular motion, with moves along the circumference of a circle (v=rω). meaning it travels around a its speed varying as it moves circular path at a constant along the path. It travels slower It is CHANGING IN It is CONSTANT IN speed. Its velocity changes at the top of the circle and DIRECTION but also in MAGNITUDE because direction continuously, but the accelerates toward the bottom, MAGNITUDE as the the object moves at a magnitude (speed) remains the moving faster there. This same throughout the motion. change in speed means both object speeds up or slows steady speed around the the direction and magnitude of down along the circular circle. However, the the red ball's velocity are path. DIRECTION OF changing, creating a VELOCITY nonuniform motion. CONTINUOUSLY Circular Motion CHANGES because the object is always moving in Non-uniform (NUCM) Uniform (UCM) a circular path. Position - The location of an object in space. The object’s position The object’s position changes at different rates changes at a steady rate — sometimes quickly, around the circle because sometimes slowly — it moves at a constant For instance, a roller For example, a car depending on its speed speed. coaster moves faster at moving at a constant the bottom of the loop and speed around a circular slower at the top, so both track still has a changing the direction and speed velocity direction as it (magnitude of 𝑣) change moves. at different points. Imagine a toy car on a It’s like a clock’s second circular track that speeds hand moving smoothly up on one side and slows around the face. down on the other. Speed (s) - The magnitude of the velocity (s = 2πrT). It CHANGES at different It STAYS THE SAME points on the circle. For around the circle. The example, it might speed object moves smoothly at up on one part and slow a CONSTANT RATE, so down on another, so 𝑠 the value of 𝑠 (found using is NOT CONSTANT as the equation above) the object goes around. doesn’t change as it goes around. Direction of Velocity - The orientation of the velocity vector, indicates the path the object is moving. It CHANGES as the It is ALWAYS CHANGING v: Linear or tangential velocity, the speed of an object object moves in a circle, as the object moves moving along a circular path. but because the speed around the circle, but it r: Radius of the circular path. varies, the direction changes SMOOTHLY and ω (Lower-case omega): Angular velocity, the rate at  which the object rotates, measured in radians per (due to change in speed). CHANGING ALL THE second TIME. Angular Velocity (ω) - The rate at which an object rotates or revolves about an axis (ω = Δθ/Δt). It VARIES. It is CONSTANT because the object, like a ceiling fan, rotates at a steady speed around its axis. Imagine a paper circle on a string (DIY paperfuge). If you pull the string faster, the circle will spin The fan blades move faster (increased angular through EQUAL ANGLES velocity). If you shorten IN EQUAL TIME the string (making the INTERVALS, so the radius smaller), the paper angular velocity remains Think of a car driving in a circle will spin slower the same as the fan circle at a constant speed. (decreased angular spins. Even though the The car’s linear velocity velocity), as it takes less fan is rotating at a remains constant in size, time to complete a constant rate, its linear but since it’s always rotation. So, the angular velocity (v=rω) at the tip changing direction, it has velocity changes based of the blades is higher a constant acceleration on how fast you pull and than at the center, towards the center. the length of the string. because linear velocity Imagine a car driving on a depends on the distance curved road, but it speeds from the center (radius). up or slows down as it turns. The car experiences linear acceleration both because it’s changing its direction and because its speed is changing. ω: Angular velocity, measured in radians per second (rad/s). Δθ: Change in angular position or angle (in radians) the object rotates through. Δt: Change in time or the time taken for the rotation. Linear Acceleration (a) - The rate at which linear velocity changes over time (a = v2/r). ac​is the centripetal acceleration. Both the SPEED AND It STAYS CONSTANT IN v is the linear (tangential) speed of the object. THE DIRECTION OF MAGNITUDE, but the r is the radius of the circular path. THE OBJECT CHANGE. DIRECTION CHANGES Centripetal acceleration is ______ proportional This is due to both the CONSTANTLY because to the square of the speed and ______ change in direction the object is always proportional to the radius of the circular path. (CENTRIPETAL moving in a circle. Even A) Directly, inversely acceleration) and the though the speed is B) Inversely, directly change in speed constant, the object still C) Directly, directly (TANGENTIAL experiences a D) Inversely, inversely acceleration). The linear CENTRIPETAL A) Directly, inversely acceleration in NUCM will acceleration (pointing have two components: toward the CENTER OF If the speed is 2 m/s and the radius of the centripetal acceleration THE CIRCLE) because circular path is 4 m, the centripetal acceleration (towards the center) and the DIRECTION OF is _____? tangential acceleration VELOCITY IS A) 0.5 m/s² B) 1 m/s² CENTRIPETAL force still The force (CENTRIPETAL C) 2 m/s² acts toward the center of force, FC = mv2/r) is D) 4 m/s² the circle to keep the CONSTANT because the B) 1 m/s². object in motion, but the speed of the object is TANGENTIAL force (𝐹𝑡​) constant. The force keeps also plays a role. This the object moving in a tangential force changes circular path by the object's speed along continuously pulling it If the speed is 3 m/s and the radius of the the path (either speeding toward the circle's center. circular path is 6 me, the centripetal acceleration it up or slowing it down). is _____? Tangential force is A) 0.5 m/s² responsible for changing B) 1.5 m/s² the object's linear velocity C) 3 m/s² (speed). It acts along the D) 9 m/s² tangent to the circular B) 1.5 m/s² path. Think of a car driving around a circular track at a constant speed. The It indicates that ________ speeds or ________ force required to keep the radii result in greater centripetal acceleration. car in the circular path A) Lower, shorter Consider a car driving stays the same because B) Higher, shorter around a banked curve on the speed does not C) Higher, larger a racetrack. As the car change. D) Lower, larger speeds up, it may B) Higher, shorter experience a change in its Angular Acceleration (α) - The rate with which its velocity along the curve angular velocity changes with time (α=Δω/Δt). (speeding up or slowing down). The centripetal The object’s speed is The object moves at a force keeps the car changing as it moves constant speed, so the moving along the curve, along the circular path, so angular velocity does not while the tangential force the angular velocity also change. Angular (due to friction between changes. Angular acceleration (α) is ZERO the tires and the road or acceleration (α) is because there is no engine power) affects how NON-ZERO because the change in the rate of fast the car is moving RATE OF ROTATION IS rotation. along that curved path. INCREASING OR DECREASING. Examples Winds in cyclones Earth's rotation on its (increasing and axis (steady rotational decreasing speed as speed) they move in circular Ionized gases in a paths) cyclotron (constant Electrons in an atom speed in a magnetic Imagine a carousel (electrons move in orbits field) spinning at a constant with varying speeds Washing machine drum Think of a figure skater speed. The riders move in depending on the atom's (spinning at a constant performing a spin. As they a circular path at a steady energy state) speed) pull in their arms, they pace, but the angular Motor shafts in A bicycle wheel rotating speed up (change in velocity doesn't change, machines (rotational (constant speed on a flat linear velocity), which so there’s no angular speed varies depending surface) causes a change in acceleration. on load) Motorbikes Wall of angular velocity (Δω). As Gymnast performing a Death (performer moves their angular velocity spin on the floor (change at a constant speed in a increases, the angular in speed as the circular path with a fixed gymnast’s body position radius) acceleration (α=Δω/Δt) is alters) non-zero and has a Rotating music discs in curved direction. a turntable (changing Force - Keeps an object moving in a circular path (FC speed in some types of = mv2/r). equipment)  Twirling Lasso Trick Why can the formula for linear velocity in Uniform (speed and radius of the Circular Motion, 𝑣=2𝜋𝑟/𝑇, be rewritten as 𝑣=2𝜋𝑟𝑓? lasso change as it's A) Because frequency (f) is the inverse of the period twirled in different (T), so 𝑓 = 1/𝑇 directions) B) Because radius (r) is inversely proportional to the velocity (v). How is UCM related to the concept of time? C) Because the speed of the object is directly It is closely related to time because it describes how proportional to the time taken to complete a an object moves in a circle at a constant speed over rotation. time. D) Because the object's velocity is constant Time is important because the object takes a throughout the motion, and this relationship does specific amount of time to complete one full not depend on frequency. rotation around the circle. This time is called A) Because frequency (f) is the inverse of the period (T), so 𝑓 = 1/𝑇 the period (T), and it’s the time it takes for the The period (T) is the time it takes for the object to complete one full rotation. Frequency (f) is the number of rotations per second, which is the inverse of the object to go around the circle once. period: 𝑓 = 1/𝑇. Substituting 𝑓 = 1/𝑇 into the formula 𝑣=2𝜋𝑟/𝑇 gives 𝑣=2𝜋𝑟/𝑇. Formula for Time in UCM: Possible Centripetal Forces The period (T) can be related to the The specific centripetal force depends on the nature of frequency (how often something the circular motion and the objects involved. happens in a given amount of time). The frequency is the number of times the object completes a full rotation per second. The formula that connects period and frequency is: T is the period (in seconds), f is the frequency (how many rotations per second). For example, if a In the equation FC = mv2/r, what happens to the wheel spins 2 centripetal force if the radius (𝑟) is doubled, while times per second, keeping the velocity constant? then the period A) The centripetal force is halved would be: B) The centripetal force is doubled This means it takes half a second to complete one full C) The centripetal force is quadrupled rotation. D) The centripetal force remains unchanged A) The centripetal force is halved Imagine you’re on a merry-go-round In the equation FC = mv2/r, how does the (carousel). As it spins, centripetal force change if the velocity (𝑣) is it takes a certain doubled, while keeping the radius constant? amount of time to A) The centripetal force is halved complete one full B) The centripetal force is doubled rotation. That time is C) The centripetal force is quadrupled your period (T). D) The centripetal force remains unchanged If the carousel completes 1 full rotation every 10 C) The centripetal force is quadrupled seconds, then its period is 10 seconds. If the mass (𝑚) of an object is increased in the If the carousel spins faster, the period decreases (it equation FC = mv2/r, what happens to the takes less time to complete one full rotation), and the centripetal force? frequency (how often the carousel spins per second) A) The centripetal force decreases increases. B) The centripetal force increases C) The centripetal force remains unchanged D) The centripetal force becomes zero B) The centripetal force increases In the equation FC = mv2/r, which of the following variables is directly proportional to the centripetal force? A) Mass (𝑚) B) Velocity (𝑣) C) Radius (𝑟) D) Both A and B D) Both A and B  Car turning a curve: When a car takes a turn on a flat road, the friction between the tires and the road keeps the car from sliding off the curve. This friction acts as the centripetal force. Spinning a coin on a table: As the coin spins in a circle, the friction between the coin and the table prevents it from sliding off, keeping it in motion. A person spinning a ball on a string: The Static Friction friction between the ball and the string It is the force that helps keep the ball moving in a circular prevents an object from path. sliding or moving when it’s in contact with a surface. Tension It is the force in a stretched In circular motion, static friction can act as the string, rope, or cable. In force that pulls an object toward the center of circular motion, tension can the circle, keeping it moving in that path. act as the centripetal force that pulls the object toward the center of the circle. FC​: Centripetal force, the force that keeps an object moving in a circular path, directed toward the center of the circle. Ff​: Frictional force that provides the centripetal force FC​: Centripetal force, the net force required to keep in cases such as an object moving in a circular an object moving in a circular path. It is directed path on a flat surface. toward the center of the circle and is measured m: Mass of the object moving in the circular path, in (N). measured in (kg). Tstring​: Tension in the string or rope that is providing v: Linear (tangential) velocity of the object, the centripetal force. This tension must be measured in (m/s). sufficient to keep the object moving in a circle. r: Radius of the circular path, measured in (m). m: Mass of the object being swung in the circular μ: Coefficient of friction, a dimensionless value that path, measured in (kg). represents the friction between two surfaces. v: Linear (tangential) velocity of the object, FN: Normal force, the force perpendicular to the measured in (m/s). surfaces in contact, usually equal to the weight r: Radius of the circular path, measured in (m). of the object if the surface is horizontal If the speed of the object __________, the ___________ speeds or ___________ radii required centripetal force __________. increase the required centripetal force. A) Increases, decreases A) Higher, shorter B) Increases, increases B) Lower, shorter C) Decreases, increases C) Higher, larger D) Decreases, decreases D) Lower, larger B) Increases, increases A) Higher, shorter A smaller radius means that the required The __________ the normal force (such as when centripetal force is ________ for the same speed. the mass of the object increases), the __________ A) Decreased the maximum frictional force that can act before B) Increased slipping occurs. C) Unchanged A) Higher, lower D) Doubled B) Higher, higher B) Increased C) Lower, higher Swinging a metal ball on a string: When D) Lower, lower you swing a metal ball tied to a string in a B) Higher, higher  circle, the tension in the string provides the r: Distance between the centers of the two masses, centripetal force that keeps the ball moving measured in (m). in that circular path. As the distance _________, the gravitational force becomes _________. A) Increases, stronger B) Decreases, weaker C) Increases, weaker D) Decreases, stronger C) Increases, weaker Merry-go-round (flying fiesta): The tension in the ropes or chains that hold the riders Planet orbiting the Sun: The gravity on a merry-go-round keeps them in a between the Earth and the Sun keeps the circular motion. Earth in orbit around the Sun, acting as the Tug-of-war game: In a tug-of-war, when the centripetal force. rope is pulled tight, tension acts to pull Moon orbiting the Earth: The Moon is held objects toward the center of the circle if in orbit around the Earth by gravity, which they are spinning around. pulls the Moon towards the Earth, providing the centripetal force. Spring Satellites in orbit: Satellites in space stay in It can provide a restoring force that circular orbit around the Earth because gravity provides the necessary centripetal pulls an object toward a center. This force can force to keep them in motion. also act as a centripetal force when an object is moving in a circular path. Mass attached to a spring: If a mass is attached to the end of a spring and spun in a circle, the spring’s restoring force acts as the centripetal force, pulling the mass toward the center. Shock absorbers in a car: The springs in a car’s shock absorbers help absorb bumps, Normal Force and they can also act to pull the car back to its central position when it moves. Fidget spinner: The spring-like tension in the spinner’s axle helps keep the spinner rotating around its center point. It is the force exerted by a surface that supports the weight of an object resting on it. It acts perpendicular (normal) to the surface. In Gravity circular motion, normal force can act as the It is the force of attraction centripetal force when an object is on a curved between two masses. In surface. circular motion, gravity can act as the centripetal force that pulls an object toward the center of the circle. FC​: Centripetal force, the force that keeps an object moving in a circular path, directed toward the center of the circle. FN​: Normal force, the force perpendicular to the surface that supports the weight of the object. It acts outward from the surface. θ: The angle of inclination of the surface with FC​: Centripetal force, the force that keeps an object respect to the horizontal. The tangent of this moving in a circular path, directed toward the angle relates the vertical height to the center of the circle. horizontal distance. FG​: Gravitational force, the attractive force between μ: Coefficient of friction, a dimensionless value two masses due to gravity. representing the frictional interaction between m1​and m2: The masses of the two objects two surfaces. It varies based on the materials in interacting with each other, measured in (kg) contact. G: Universal gravitational constant As the θ (the angle of inclination) ____________, speed, but it constantly alters its direction to keep it the component of the gravitational force acting moving along the curve. It is always perpendicular to parallel to the surface ____________, affecting the object's linear velocity vector (which is tangent to both the normal force and the centripetal force the circle). required. A) Increases, increases B) Increases, decreases C) Decreases, increases D) Decreases, decreases A) Increases, increases Car on a banked curve: When a car turns on a banked curve, the normal force from the road can provide the centripetal force that keeps the car moving in a circle. In a circular motion, CENTRIPETAL FORCE is the force that acts on an object moving in a circular path to keep it in that path. Like centripetal acceleration, centripetal force is DIRECTED TOWARD THE CENTER of the circle along the radius. Roller coaster: The normal force from the tracks on a roller coaster provides the centripetal force to keep the coaster moving along its curved path. A ball in a bowl: When a ball is rolled in a bowl, the normal force from the bowl’s surface pulls the ball towards the center, acting as the centripetal force. Is projectile motion related to circular motion? Projectile Motion Circular Motion Similarities Both involve accelerated In curved paths, both motion: Projectile motion exhibit forces that change involves vertical the direction of motion, acceleration due to with gravity affecting gravity, while circular projectiles and centripetal In circular motion, the LINEAR VELOCITY VECTOR is motion involves force acting in circular ALWAYS TANGENT to the circle at the point of the centripetal acceleration. motion. object's position. It is PERPENDICULAR TO THE Differences RADIUS of the circle, which connects the center of the Involves an object moving An object moves in a circle to the object's position. along a curved path due circular path, constantly to its initial velocity and changing direction to gravity acting vertically maintain a fixed radius. downward. The motion The object experiences has two components: centripetal acceleration horizontal (constant toward the center of the velocity) and vertical circle. (accelerated due to gravity). Escape Velocity The minimum speed an object must have to break free In a circular motion, CENTRIPETAL ACCELERATION from a planet or celestial body’s gravitational pull is directed TOWARD THE CENTER of the circle, without further propulsion. It is independent of the ALONG THE RADIUS. It is responsible for changing object’s mass but depends on the mass and radius of the direction of the object's velocity as it moves along the planet or body being escaped from. the circular path. It does not change the object's Orbital Velocity counteract the pull of gravity, preventing the water The velocity an object must have to enter a stable from falling out, even when the bucket is upside down! orbit around a planet or celestial body. Orbital velocity depends on the mass of the central body and the Centripetal Force radius of the orbit. The force that pulls an object toward the center of a circular path keeps it moving in a circle. It doesn't change the object's speed but changes its direction. Why Does a Candle Flame Point Toward the Center When Spun? When a candle is spun, the flame points toward the center because of INERTIA and CENTRIPETAL FORCE. The flame's gases want to move in a straight line (inertia), but the spinning motion pulls them inward Escape Velocity is the speed needed to break free toward the center, making the flame lean toward the from the gravitational pull, while Orbital Velocity is the middle. speed needed to remain in a stable orbit. Escape Watch this for reference: velocity is always greater than orbital velocity for the https://www.youtube.com/watch?v=qPyz1MwZSFs same distance from the planet. How is the 2nd Law of Motion related to Circular Is there inertia in circular motion? Motion? Inertia is the tendency of an object to resist changes The second law of motion, F = ma (Force equals mass in its state of motion (Newton’s 1st Law of Motion). In times acceleration), is directly related to circular circular motion, inertia is present because the motion because when an object moves in a circle, it OBJECT STILL WANTS TO MOVE IN A STRAIGHT experiences centripetal acceleration, even though its LINE DUE TO ITS VELOCITY. However, the speed may be constant. centripetal force continually redirects the object toward The object accelerates toward the center of the circle's center, causing it to follow a circular path the circle, CHANGING ITS DIRECTION rather than moving in a straight line. continuously. This acceleration is called centripetal acceleration, and it's always DIRECTED INWARD. According to Newton's second law, THIS INWARD ACCELERATION REQUIRES A FORCE, known as centripetal force. The magnitude of this force is given by F = ma, where m is the object's mass and a is the centripetal acceleration. The centripetal force is _______ proportional to the mass of the object. A ______________ object requires a greater centripetal force to maintain the same circular motion. A) Directly; heavier B) Inversely; lighter C) Directly; lighter D) Inversely; heavier A) Directly; heavier Centripetal force is directly proportional to the mass of the object. A heavier object requires a greater centripetal force to maintain the same circular motion. The centripetal force is ________ proportional to the square of the velocity. Doubling the velocity Why doesn't the water fall out of a bucket when it's requires ________ times the centripetal force to swung in a circular motion? maintain circular motion. The water stays in the bucket due to a combination of A) Directly; 2 CENTRIPETAL FORCE and INERTIA. When the B) Inversely; 2 bucket is swung in a circle, the water's inertia (its C) Directly; 4 tendency to stay in motion) tries to keep it moving in a D) Inversely; 4 straight line. However, the bucket provides an inward C) Directly; 4 The centripetal force is directly proportional to the square of the velocity. If you double force, pulling the water toward the center of the the velocity, the centripetal force increases by a factor of 4 (since 𝐹∝𝑣2). circular path. As long as the bucket is moving fast enough, the centripetal force is strong enough to The centripetal force is ________ proportional to Reaction: According to Newton's Third Law, the riders the radius of the circular path. As the radius exert an equal and opposite force on the wall, which _________, the required centripetal force is an outward force. This is the riders' inertial force decreases. due to their tendency to move in a straight line. Since A) Directly; increases they are moving in a circular path, the outward force B) Inversely; increases they exert on the wall is balanced by the inward C) Directly; decreases centripetal force that the wall applies to them. D) Inversely; decreases B) Inversely; increases Is there really what we call a centrifugal force? The centripetal force is inversely proportional to the radius of the circular path. As the radius increases, the required centripetal force decreases. How is the 3rd Law of Motion related? The third law of motion, "For every action, there is an equal and opposite reaction," is related to circular motion because it explains the forces involved between objects in motion. Action: The object (e.g., a car, a satellite, or the bucket in the bucket-and-water example) exerts an inward force toward the center of the circular path. This is the centripetal force that keeps the object in motion along the circular path. Reaction: The object being acted upon (like the water in the bucket) exerts an outward force (called centrifugal force, although not a real force in physics, it describes the apparent effect) that seems to push it away from the center, which is equal in magnitude but opposite in direction to the centripetal force. Why learn about UCM? In the Gravitron ride (a popular carnival ride), the 1. Understanding Real-World Motion (BUT WHY action and reaction forces are explained by Newton's & SPECIFICALLY HOW?) Third Law of Motion. 2. Foundation for Advanced Physics (BUT WHY Action: The wall of the & SPECIFICALLY HOW?) Gravitron pushes inward on 3. Applications in Engineering and Technology the riders with a centripetal (BUT WHY & SPECIFICALLY HOW?) force. This force is what keeps the riders pressed against the wall as the ride spins in a circle. This is the force that makes the riders feel like they're stuck to the wall, even though the floor drops away.