Rotational Motion

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Indiana University, Bloomington *

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Physics

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Apr 3, 2024

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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 .

Rotational Kinetic Energy 𝐾𝐾𝐾𝐾 = 1 𝐼𝐼𝜔𝜔 2 2 + 1 𝑚𝑚𝑣𝑣 2 2 (5) A rotating object has kinetic energy that is the sum of its rotational energy ( 1 𝐼𝐼𝜔𝜔 2 ) and, if it is moving 2 (like a hoop rolling down an incline), its translational energy ( 1 𝑚𝑚𝑣𝑣 2 ). Since it is a sum, rotational energy 2 has the same units as regular kinetic energy: Joules . Torque τ Torque: the rotational equivalent of force; the “twist” that changes the direction of rotation; the straw that stirs the drink. Torque is given by a rotation equivalent of Newton’s Second Law ( F = ma ): τ = Iα = rF sin θ (6) where θ is the angle between linear r and F (see Figure 3). Torque is measured in Netwons*meters . More force means more torque, and a greater distance from the reference point also means more torque. The torque is biggest when the angle between force and distance is maximized at 90° (see Figure 3). Figure 3: Right-hand rule for torque.

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.

Procedure & Study Guide Submit only this portion of the document on Canvas, leaving off all of the introductory pages. A. Rotating Platform Theory: Refer to the sections on Torque, Conservation of Momentum, and Angular Momentum (Conservation) in the Introduction. The rotating platform allows rotation about the vertical axis, which means that there are no external torques acting in a vertical direction. Thus, the vertical component of L is conserved. Apparatus: The apparatus consists of a simple rotating platform. Begin each portion with one lab partner sitting on the platform. When necessary, the other lab partner tosses the medicine ball or passes the spinning bicycle wheel. Safety Notes: Have a person ready to steady you while standing on the rotating platforms. Do not extend your hands out too quickly when you have weights in your hand. You could inadvertently hit someone with them. Do not drop the bicycle wheels or stand on the bicycle wheels. They are not unicycles. At best you will break the wheel. At worst, someone could be seriously injured. Figure 6: Rotating platform. Questions: 1. Sit on the platform with your arms outstretched. Make sure no one is in the way and have your partner spin you. Pull your arms in. Why do you speed up? What happens if you do this with weights in your hands? Hint: Consider conservation of angular momentum and moment of inertia. 2. Sit on the platform and take the spinning bicycle wheel from your partner, who hands it to you with the axis horizontal. Explain what happens when you turn the wheel to hold the axis vertically. Hint: Consider conservation of angular momentum along the vertical axis.

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