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Lab 11 – Torque and Rotational Inertia
THEORY:
 
Torque is the rotational analogue to force and is responsible for why an object’s rotational motion changes. The definition of torque is
 
 
The magnitude of the torque is given by  where is the angle between the force direction and the vector that points from the axis of rotation to the point where the force is applied. For a non-zero Torque you need:  .
 
A rigid body that has a non-zero net torque applied to it will experience a non-zero angular acceleration according to Newton’s 2nd Law of Rotational Motion, given by
 
where I is the Rotational Inertia or Moment of Inertia of the rigid body; the rotational analogue of mass for translational motion and is the angular acceleration. The angular acceleration results in a change in the angular velocity of the object. If the torque is constant, then the angular acceleration is constant and the angular velocity changes with time according to
 
and the angular displacement changes according to
 
The rotational inertia of an object depends on its mass and how the mass is distributed about a defined rotational axis. For certain geometric objects the rotational inertia about an axis through the center of mass can be determine by integration. For a hoop or thin-walled cylindrical shell of mass M and radius R, the rotational inertia about an axis through its CM and normal to its face is given by
 
 
For a solid disk or solid cylinder of mass M and radius R , the rotational inertia about an axis through the CM and normal to its face  is given by
For a solid sphere of mass M and radius R , the rotational inertia about an axis through its CM is given by
 
For example, a mass m is suspended by a string wrapped around a solid disk or solid cylinder as in Figure 1, would exert a torque about an axis through the disk’s center of mass causing the disk to rotate.
 
 
 
 
 
 
 

 
 
 
 
Figure 1. A mass m suspended from a solid cylinder (disk) of mass M and radius R exerts a torque.
 
Rotational kinetic energy is also associated with rotating objects  is given by  which is the rotational analogue of .
If more than one torque act on a rigid body, the net torque is vector sum of the torques. If all the forces and radial distances for the axis of rotation are in a single plane, then torques can be given positive or negative values depending on whether they would result in CCW rotations (+) or CW rotations (-). In Figure 2, two masses are suspended from a solid disk connected by a string wrapped around the outside of the disk. In this case, the tensions are different on the different sides, resulting in a non-zero net torque about rotational axis and an angular acceleration for the disk.
 
 
 

 
 
 
 
 
 
 
 
 
Figure 2. Two masses suspended from a solid cylinder

  1. Initial Calculations:

 

  1. For the setup shown in figure 1, the suspended mass is and the pulley is a thin-walled cylindrical shell of mass   and radius . Set up a free body diagram for the suspended mass labelling the forces acting on the mass m and using Newton’s second law of linear motion, write and expression for the acceleration of the mass in terms of the tension in the string and the mass.

 
 
 
 
 
 
 

  1. For the same setup, using Newton’s 2nd Law for rotational motion, write down an expression for the angular acceleration of the pulley in terms of its mass and radius and the tension in the string.

 
 
 
 
 
 

  1. Assuming the tension in part (a) is the same as in part (b), determine the angular acceleration of the pulley, the linear acceleration of the suspended mass and the string that is connected to it, and the tension in the string.

 
 
 
 
 
 
 
 
 

  1. Determine the magnitude of the torque acting on the pulley in problem above.

 
 

  1. Consider the setup in Figure 2, with the left suspended mass is the right suspended mass is  , and the pulley is a thin-walled cylindrical shell of mass   and radius . Set up free-body diagrams for each suspended mass labelling the forces acting on the mass.  For each free-body diagram, use Newton’s second law of linear motion to write an expression for the acceleration of the mass in terms of the tension in the string on that side of the pulley and the mass.

 
 
 
 
 
 

  1. For the same setup, using Newton’s 2nd Law for rotational motion, write down an expression for the angular acceleration of the pulley in terms of its mass and radius and the tension in the string.

 
 
 
 
 

  1. Simultaneously solving the two equations in part (e) and the equation in part (f), determine the angular acceleration of the pulley, the linear acceleration of the suspended masses and the string that is connected to it, and the tensions in the string on each side of the pulley.

 
 
 
 
 
 
 
 

  1. Determine the magnitude of the torque acting on the pulley in problem above.

 

  1. Lab Activity:

Open the following link for a simulation of the rotation motion lab:  https://ophysics.com/r5.html

  1. Select the “Falling Mass” simulation and set the initial parameters for the simulation of Run the simulation and observe the values for the net Torque on the pulley, the acceleration of the mass, and the angular acceleration of the pulley. Compare your observations to your calculations in parts 1a-d. If the magnitudes of the values do not match closely, review your calculations in the previous section.

 
 
 

  1. Is the arithmetic sign of the torque and angular velocity in agreement with the convention that we are using in the textbook?

 
 
 

  1. Repeat the simulation for a solid cylinder and for a solid sphere using the same radius and mass of the rotating object and the same falling mass. Which object (cylindrical shell, solid cylinder, or solid sphere) has the largest angular acceleration. Determine for each rotating object (cylindrical shell, solid cylinder and solid sphere) the rotational inertia about the axis through the center of mass. Use this in explaining the results for the angular acceleration of the pulley in each case.

 
 
 
 
 
 

  1. For this simulation, let’s see if the timing is correct. Set the rotating object back to the cylindrical shell. Using a stop watch on your phone or watch, measure the time it takes the cylindrical shell to rotate one-half revolution (radians). Given the angular acceleration of the pulley and the kinematic expressions for constant angular acceleration, determine the time it should have taken for the angular displacement to be Is the simulation in real time? Or in slow motion?

 
 
 
 
 

  1. Select the “Two Masses” simulation, set the initial parameters to be left suspended mass is the right suspended mass is  ,  and the pulley is a thin-walled cylindrical shell of mass   and radius . Run the simulation and compare the resulting magnitude of the net torque, angular acceleration of the pulley and linear acceleration of the masses with your calculations from 1.e-g.

 
 
 

  1. Reflection on the result
  2. For the case in Activity 2a, the cylindrical shell rotates through one-half revolution during the run. Given the radius of the pulley, how far as the suspended mass dropped? Determine the potential energy change for the system.

 
 
 
 
 

  1. Determine the time interval it takes for the pulley to rotate that half-revolution (from the kinematics of the rotation). Determine the final total kinetic energy of the system of the pulley and the suspended mass at the end of the run. Determine the change in the kinetic energy of the system.

 
 
 
 
 
 

  1. Is the total mechanical energy of the system conserved in this simulation? Justify your answer.

 
 
 
 
Conclusion: