08-Rotational Dynamics-PHYS 2425-Handout and Lab Report V2 (1)

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2425

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Dec 6, 2023

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ROTATIONAL DYNAMICS NOTE: You are expected to directly put your answers in this document to be uploaded for submission onto the appropriate dropbox in your Physics Lab Blackboard course. Make sure you format your answers such that the instructor will know these are your responses to the question. Objectives: In this lab you will be studying the factors affecting torque, moment of inertia, and angular momentum. Materials: Rotational Dynamics PHET Simulation PART I. TORQUE Just as forces produce linear motion, torques produce rotational motion. A torque occurs when a force F acts on an extended object at some distance r from its center of mass as in the figure shown at left. The torque 𝛕 produced by the force F is given by 𝛕 → = 𝐫 → 𝑥𝐅 → The magnitude of the torque τ is equal to the product of the force’s magnitude F and the perpendicular distance rsinθ (referred to as moment arm) from the axis of rotation to the position of the force. τ = rF sinθ where θ is the smaller angle between r and F. Just as forces produce acceleration of objects, A net torque produces an angular acceleration on objects. Compare the formulas below S F = ma ® S τ = Ia where I is the moment of inertia (our rotational analogue of mass, discussed in Part 2) and a is the angular acceleration that results from a net torque. In this lab we will be working with “ 1 Dimensional ” torques, and so motion as well as angular acceleration will be described as having either counterclockwise (+) direction, or clockwise (-) direction. PROCEDURE: Preliminaries: 1. Install the Java software in your computer. The following link provides the versions of Java for different operating systems: https://www.java.com/en/download/manual.jsp

2. Download the Rotational Dynamics PHET Simulation: https://phet.colorado.edu/en/simulation/legacy/torque A. Variation of Torque and distance with Constant Force 1. Click on the Torque Tab in the PHET Simulation as shown in the Figure. The picture below shows the initial settings of the animation. Figure 1.2. Initial Settings of the Animation

2. Set the Force equal to 5N by clicking the white box under Applied Force and typing 5 as shown in the figure at above. 3. Set the distance (r) equal to 1. Do this by clicking on the white box under Radius of Force and typing 1. 4. Click on any of the Go icon and let it run for about 3 seconds before clicking stop. Read off the value of Net Torque as indicated on the graph. 5. Complete Table 1 by repeating steps 3 and 4 for the different values of r as indicated. Table 1: Relationship between distance and Torque with the Force constant. Force (N) Distance (m) Torque 5 1.0 5 5 1.5 7.5 5 2.0 10 5 2.5 12.5 5 3.0 15 5 3.5 17.5 5 4.0 20 6. Plot the Torque (y axis) vs. distance (x-axis) using Excel or Capstone, and draw a best-fit curve. Take a screen shot of the graph and paste it below: [You can download a 60-day trial version of Capstone from: https://www.pasco.com/downloads/capstone ]

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Question 1: What is the relationship between the Torque and the distance of the force from the axis of rotation? The functional relationship between the torque and the distance is a linear equation B. Variation of Torque and Force with the distance constant In this part, you will set the distance of the force from the axis of rotation to be 4 m. Complete Table 2 by varying the force as indicated. Table 2: Relationship between Torque and Force with distance constant. Distance(m) Force (N) Torque 4 1.0 4 4 1.5 6 4 2.0 8 4 2.5 10 4 3.0 12

7. Plot the Torque (y axis) vs. Force (x-axis) and draw a best-fit curve. Take a screen shot of the graph and paste it below. Question 2: What is the relationship between the Torque and the Force when the distance of the force from the axis of rotation constant? The functional relationship between the torque and the force when the distance is constant is a linear relationship. Question 3: What is the mathematical relationship between the Torque, Force, and distance of the force from the axis of rotation based on your results in parts A and B of the experiment. The mathematical relationship between the torque, force, and the distance of the force is that to get the torque simply multiply the force and the distance. As the distance increases, the torque increases much more significantly than when the force is increased based on both trials.

C. Net Torque 1. Change the “Force of Brake” to 4.0 N, the applied Force to 4 N and the “Radius of Force” to 4.0 m. Click GO! and let the simulation run for about 3 seconds before clicking stop. Question 4; What is the torque produced by the Force of the Brake? The torque produced is -16 N. Question 5: What is the torque produced by the applied Force? The torque produced is 16 N. Question 6: What is the net torque? The net torque is 0 N. Question 7: Describe the motion of the platform. The platform is spinning slowly as the brake is being applied. 2. Change the “Force of Brake” to 3.0 N while keeping the applied Force and the “Radius of Force” the same as in step 2. Click GO! Click stop after about 3 seconds. Question 8; What is the torque produced by the Force of the Brake? The torque produced is -12 N. Question 9: What is the torque produced by the applied Force? The torque produced is 16 N. Question 10: What is the net torque? The torque produced is 4 N. Question 11: Describe the motion of the platform. In which way does it rotate (answer: clockwise or counterclockwise)? The platform is moving faster than the previous trial. It is moving counterclockwise.

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Question 12: How do we calculate the net torque? Adding the Applied Torque to the Torque of Brake, which is normally a negative number. Part II. Torque and Moment of Inertia. The moment of inertia (I) of an object depends on the masses of the particles making up the object and their distances from the rotation axis. The moment of inertia depends on how the masses are distributed with respect to the rotation axis. The moment of inertia of rigid objects is determined by using the following equation: 𝐼 = ∫ r 2 ρdV Using the above equation, we can derive the following moment of inertia of homogeneous (uniform mass distribution) rigid bodies. Object Geometry Moment of Inertia (about the center of mass) Hoop or Thin Cylindrical Shell I = MR 2 Hollow Cylinder I = 1 2 M(R 1 2 + R 2 2 ) Solid Cylinder or disk 1 2 𝑀𝑅 2 Solid Sphere 2 5 𝑀𝑅 2 A. Moment of Inertia 3. Click on the Moment of Inertia tab in the simulation. Set the values of the quantities as follows: Torque = 0.0 N.m R (outer radius) = 4.0 m R (inner radius) = 0.0 m Platform mass = 0.20 kg Force of Brake = 0 N Angle units: Set to radians

4. Click Go! and let the simulation run for about 3 seconds before clicking stop. Read off the moment of inertia from the appropriate graph window. Record it in Table 3. 5. Repeat the above steps for the inner radius indicated in Table 3. Table 3. Moment of Inertia Outer Radius (m) Inner radius (m) Moment of Inertia (kg.m 2 ) 4.0 0.0 1.6 4.0 0.5 1.625 4.0 1.0 1.7 4.0 1.5 1.825 4.0 2.0 2.0 Question 13: What happens to the moment of inertia when the inner radius was increased? Explain why this is so . The moment of inertia increased slightly, but increased, nonetheless. This can be so because the rotation axis, or the inner radius, is larger. The mass is concentrated at the rim and is further away from the center, thus the radius is increased. B. Variation of Angular Acceleration with Moment of Inertia (with constant Torque) 1. Set the values of the quantities as follows: Torque = 2.0 N.m R (outer radius) = 4.0 m R (inner radius) = 0.0 m Platform mass = 0.05 kg Force of Brake = 0 N Angle units: Set to radians 2. Click Go! Record the moment of inertia and angular acceleration in Table 4. 3. Repeat Steps 1 and 2 by changing the mass of the platform as indicated in Table 4. Keep the other quantities the same.

Table 4: Relationship between angular acceleration and moment of inertia with constant Torque Torque (N.m) Platform Mass (kg) Moment of Inertia, I (kg.m 2 ) Angular Acceleration, α (rad/s 2 ) 2 0.02 0.16 12.5 2 0.04 0.32 6.25 2 0.08 0.64 3.125 2 0.10 0.8 2.5 2 0.12 0.96 2.083 2 0.14 1.12 1.786 2 0.16 1.28 1.562 2 0.18 1.44 1.389 2 0.20 1.6 1.25 2 0.22 1.76 1.136 4. Generate a plot of the angular acceleration, α (y-axis) versus 𝟏 𝑴????𝒕??𝑰??𝒓𝒕𝒊𝒂 (x-axis). Draw a best-fit line and determine the slope. Take a screen shot of the graph and paste it below.

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Question 14: What is the slope of the α graph vs. 𝟏 ?????𝒕??𝒊?𝒕?𝒓𝒕𝒊𝒂 ? The slope is 1.997. Question 15: What is the relationship of the torque, moment of inertia and the angular acceleration? The functional relationship between the moment of inertia and angular acceleration is linear. The torque is equal to the moment of inertia times the angular acceleration. As the moment of inertia increases, the angular acceleration decreases. Group Contributions : Briefly state the name and contribution of each lab member during this experiment 1. Name – Dyana De Leon Contribution – Done individually ; Helped each other throughout 2. Name – Ruby Lopez Contribution – Done individually ; Helped each other throughout