Rotational Motion

Grading Rotational Motion This lab is a “check lab”. To receive full credit, record your observations, make sketches, answer the questions for each of the stations, and submit the completed questions to Canvas. Although your Instructor will check your answers to make sure you understand the physics, you will not be graded on it. The goal of the lab is to help you understand some aspects of polarization. If you make an effort to do this and participate significantly, you will receive 5 points for the lab. Important Safety Note: When used properly, the apparatus in this lab is quite safe. However, when used improperly some of the equipment has the potential to cause serious personal injury. Please adhere to the safety precautions given in each section. Any student who fails to follow the above rules will be expelled from the lab and will receive a grade of zero for the lab. Introduction In this lab you get "hands on" experience with rotational motion and the seemingly mysterious concepts (as perceived by many students) of angular motion. This is a qualitative lab and is best understood in qualitative terms: greater than, less than, faster, lighter, etc. It emphasizes the vector nature of rotational motion and relies heavily on the right-hand rule to determine the direction of the vectors. The Introduction lays out important definitions and concepts of rotational motion. Read the entire Introduction before coming to lab and refer back to the summary during the lab. It will save you some time completing the questions! Angular Position θ Angular position : a measure of displacement in terms of angle θ . The arc length s is the distance along the circumference of a circle. It is given by: s = rθ (1) where r is the radius of the circle transcribing the arc and θ is the angular distance from the zero-angle position. When using this relationship, one should remember to use radians for the angular measure rather than degrees. Figure 1: Angular position. A radian is just the ratio between the radius of a circle and its diameter. Using this definition, it is relatively easy to figure out there are 2π radians in a full circle. Being the ratio of two lengths, a radian has no actual units.

Traditionally most physicists describe an angular position that increases in the counterclockwise direction of rotation. However, this is not universal. It is always good to show in a drawing which direction of rotation is considered positive. Angular Velocity ω Angular velocity : a measure of how fast a body rotates, given by: 𝜔𝜔 = ∆𝜃𝜃 = 𝑣𝑣 (2) ∆𝑡𝑡 𝑟𝑟 The first statement of the equation is angular (in terms of θ and t ), and the second is tangential (in terms of linear v and radius r ). ω is measured in radians/second . Figure 2: Right-hand rule for angular velocity. Angular velocity is a vector. To find its direction, use the right-hand rule for angular velocity. Using your right hand, curl your fingers in the direction of rotation, and ω points in the direction of your thumb (see Figure 2). Angular Acceleration α Angular acceleration : a measure of how fast a body changes its angular velocity, given by: 𝛼𝛼 = ∆𝜔𝜔 = 𝑎𝑎 (3) ∆𝑡𝑡 𝑟𝑟 This is a rate of change, just like linear acceleration a . The units are θ radians/second 2 . Angular acceleration is also a vector, and has its direction defined by ω (the right-hand rule) :  α is in the same direction as ω if the rotation of the body is speeding up (if ω is increasing)  α is in the opposite direction as ω if the rotation of the body is slowing down (if ω is decreasing) Moment of Inertia I Moment of inertia : the rotational equivalent of mass, showing how difficult it is to get an object to rotate, given by: 𝐼𝐼 = ∑ 𝑚𝑚𝑟𝑟 2 (4) Notice that the moment of inertial depends not only on mass, but more importantly on the distance each particle of mass is from the center of rotation. Moment of inertia is not a vector and has units of kilogram*meters 2 .

Angular Momentum l Angular momentum: the angular equivalent of linear momentum; a measure of an object’s rotational motion around a chosen center point. This is given by the rotational equivalent of p = mv : L = I ω = mrv sin θ (7) Angular momentum is dependent on the distribution of mass around the center point (given by I) and its rate of rotation (given by ω) . Its linear equivalent depends upon linear momentum and r ⊥ , the shortest distance between the center point and the line along which the velocity lies (Figure 4). It is measured in (kilograms*meters 2 )/second . Figure 4: Angular momentum is defined relative to some reference, or center, point. Conservation of Angular Momentum: If there are no external torques on a system, the angular momentum of the system is conserved. Many of the experiments are designed with no (or very little) external torque, so conservation of angular momentum is applicable in these situations. This means the initial angular momentum equals the final angular momentum: L i = L f , no matter what changes within the system. For instance, if I increases, then ω must decrease to conserve L . Change of Angular Momentum with Torque: If an external torque is applied to a system for a short period of time Δ t , the torque changes the angular momentum of the system: 𝜏𝜏 = ∆𝐿𝐿 ∆𝑡𝑡 The new angular momentum after time Δ t is L f =L i + Δ L , or: (8) L f =L i + τ Δ t (9) This is a vector addition. Let’s consider two cases found in this lab. Torque acts parallel to L: The head-to-tail addition of vectors yields an L f in the same direction as L i , but the magnitude is either bigger or smaller depending on whether τ is in the same direction as Li or the opposite direction. Accelerating a wheel is a good example of this case. Torque acts perpendicular to L: The head-to-tail vector addition now changes the direction of Lf , as shown in Figure 5. If Δ t is small, then Δ L is essentially the arc of the circle, so Lf = Li in magnitude. If the torque acts over a long period of time, it is analyzed by breaking time up into small Δ t and repeating the vector addition at each interval. The result is the angular momentum having the same magnitude, but it constantly changes direction. This is called precession . Figure 5: Angular momentum is defined relative to some reference, or center, point.

Summary The table below should help you keep track of some of the rotational quantities and the equations used to define them. This is not a substitution for a careful reading of the Introduction, which is required to understand the concepts! Quantity Equation Linear Equivalent Notes Angular Position θ s = rθ Position x Angular Velocity ω ∆𝜃𝜃 𝜔𝜔 = ∆𝑡𝑡 Velocity v The linear equation is: ω = v/r Angular Acceleration α ∆𝜔𝜔 𝛼𝛼 = ∆𝑡𝑡 Acceleration a Moment of Inertia I 𝐼𝐼 = � 𝑚𝑚𝑟𝑟 2 Mass m The distribution of mass defines I. Rotational Kinetic Energy 𝐾𝐾𝐾𝐾 = 1 𝐼𝐼𝜔𝜔 2 + 1 𝑚𝑚𝑣𝑣 2 2 2 Linear KE Sum of rotational and translational energy. Torque τ τ = I α Force F Angular Momentum L L = I ω Linear Momentum p Conserved in absence of torque: L i =L f Apparatus This lab consists of 6 stations, each with their own apparatus. Refer to the Apparatus section for each station in the Procedure & Study Guide.

3. Sit still on the platform and take the spinning wheel from your partner, who this time hands it to you with the axis vertical. Hold the shaft with one hand and use your other hand to grab the wheel to stop it. Explain what happens. Hint: Conservation of angular momentum and moment of inertia.

ℎ𝑜𝑜𝑜𝑜𝑜𝑜 B. Angular Momentum Races Theory: Refer to the sections on Rotational Kinetic Energy and the Moment of Inertia in the Introduction. If the object starts at the top of an incline and rolls down, the kinetic energy equals the potential energy lost as it drops through a vertical height h : Rewriting this using ω = v / r , 1 𝐼𝐼𝜔𝜔 2 1 2 𝑣𝑣 1 + 𝑚𝑚𝑣𝑣 2 2 2 1 = 𝑚𝑚𝑚𝑚ℎ 2 𝐼𝐼 � � 2 𝑟𝑟 + 𝑚𝑚𝑣𝑣 2 = 𝑚𝑚𝑚𝑚ℎ (10) From this equation and a known expression for I , an objects velocity can be found as it rolls. Three types of objects are provided, each with their own moments of inertia and velocity:  Hoops ( 𝐼𝐼 ℎ𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑚𝑚𝑟𝑟 2 , 𝑣𝑣 2 = 𝑚𝑚ℎ )  Solid cylinders ( 𝐼𝐼 1 2 , 𝑣𝑣 2 4 ) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑟𝑟 = 2 𝑚𝑚𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑟𝑟 = 3 𝑚𝑚ℎ  Solid spheres ( 𝐼𝐼 2 2 , 𝑣𝑣 2 = 10 𝑚𝑚ℎ ) 𝑠𝑠𝑜𝑜ℎ𝑐𝑐𝑟𝑟𝑐𝑐 = 5 𝑚𝑚𝑟𝑟 𝑐𝑐𝑐𝑐𝑠𝑠𝑑𝑑 7 Apparatus: The apparatus for racing objects is a simple wooden ramp. Hold two objects still at the top of the ramp and release them at the same time to see which travels faster. Also at the station are two red and blue batons of the same mass. These are to be held, not raced on the ramp. Figure 7: Angular momentum races. Questions: 1. Objects of the same shape should have the same velocity v . Race two objects of the same shape. Is the angular velocity ω the same for each object? 2. Run races between dissimilar objects (sphere against hoop, etc.) Which is the fastest? Why? Hint: Consider the conservation of rotational kinetic energy and moment of inertia of each object.

C. Air Table Theory: Refer to the section on Angular Momentum in the Introduction. Apparatus: On an air table, plastic pucks float on a cushion of air so friction is largely eliminated. The center post serves as a fixed reference point. The pucks stick together while colliding to show the rotational interaction between them. Figure 8: Air table. Questions about the Center Post: 1. Send the puck across the table so it passes the center post at some distance r . Explain why it has angular momentum with respect to the center post even though it is going in a straight line. Hint: See Figure 4. 2. Float the spinning puck slowly over to a stationary puck that isn’t spinning so they stick together. Discuss the angular momentum before and after the collision. 3. Set both pucks spinning and glide them together so they stick. In particular, set up a situation where both pucks are initially spinning, but after collision they come to a dead stop. Discuss the angular momentum for this condition. 4. Send the pucks together on a collision course with neither one initially spinning. Unless the collision is exactly head-on, they will wind up spinning around their point of attachment. Why?

D. Whirlygig Apparatus: A ball is hooked to a string which runs through a hole and fastens to a weight mg . Place some mass on the hanger and spin the ball. Theory: Refer to the sections on Torque and Angular Momentum (Conservation) in the Introduction. In this case, the ball has an angular momentum constrained by a central force. When set in motion in an orbit, the ball experiences a centripetal acceleration: 𝑣𝑣 2 |𝑎𝑎| = 𝑟𝑟 due to the horizontal component of the tension T = mg on the string. Safety Note: When thrown, the whirlygig ball will rotate in a much larger circle than you might expect. Observe the potential path of the ball and make sure people and extraneous objects are well out of the path before throwing the ball. Figure 9: Whirlygig. Questions: 1. Does the force mg produce a torque? How does the torque affect L ? 2. As it spins, the ball stays at approximately the same radius for a few revolutions, but undeniably its radius eventually decreases as air friction takes its toll. What is the direction of the torque produced by the air friction? How does it affect L ? 3. Does the velocity v increase, decrease, or stay the same as the radius decreases?

3. Now hold the axle horizontally and again spin the wheel. What is the direction of L ? When you release the axle, what is the direction of τ ? A short time Δ t later, what is the direction of the new angular momentum L f = L i + τ Δ t ? How does this explain the precession of the wheel?

F. Air Gyro Theory: Refer to the section on Angular Momentum in the Introduction. Apparatus: The air gyro is a heavy ball with a shaft attached to it. It floats in a cup on a cushion of air. With no weight on the shaft, spin the ball and point the shaft horizontally. It precesses because there is a torque on it caused by gravity acting at the center of mass of the ball. The center of mass is not at the center of rotation. To find out where the center of mass is with respect to the center of rotation, stop the ball and hold the shaft horizontal, then release it. A weight can be added to the shaft to change the center of mass of the system. Safety Note: The air gyro is extremely heavy. It should never be removed from its base. When turning on air to the gyro, open the valve very slowly. Only a small amount of air is needed to lift the gyro. Excessive air flow creates noise and produces a very thin jet of air that could injure someone if the gyro is removed from its base. Figure 12: Air gyro. Left: Without the weight. Right: With the weight used for questions 5 and 6. Questions: 1. With no weight on the shaft, where is the center of mass with respect to the center of rotation? 2. Spin the gyro with the shaft pointing horizontal. What is the direction of L ? A short time Δ t later, what is the direction of the new L f = L i + τ Δ t ? 3. While the gyro is spinning and precessing, slip the weight on the end of the shaft. Why does this change the direction of precession?