Rotational Inertia and Dynamics
1. Consider a disc and hoop both of the same mass M, radius R and thickness t.
a) Explain why one of these objects has a larger moment of inertia (about an axis through the center of mass and perpendicular to the plane of the object) than the other. What effect does the thickness t have on the rotational inertia?
b) Explain how the rotational inertia of the disc may be obtained by adding the rotational inertias due to a series of hoops. Write down an expression (integral) representing this sum over hoop inertias.
c) Consider the rotational inertia of a hoop about an axis through its center of mass but lying in the plane of the hoop. Write down an expression for the rotational inertia of the hoop about this axis. Do you expect this expression to be greater than or less than MR2?
d) Consider the rotational inertia of the disc about the same axis as in the previous question (c). Explain how the rotational inertia can be obtained by using one of three different methods: two of these employing rods, and the other hoops.
2. Consider a pulley of rotational inertia I, and mass M with a cord wrapped around it attached to a mass m as shown in the figure. The pulley has outer radius R and the cord is wrapped around a spindle of radius r.
a) Draw free-body diagrams for both the pulley and mass m, and establish a coordinate system for the mass m and a positive sense of rotation for the pulley.
b) For each object write down equations which express Newton’s second law.
c) Assuming that m, M, and I are given, how many unknowns do the equations of part (b) involve? Can the accelerations be determined on the basis of these equations alone?
d) How is the linear velocity of the descending mass related to the angular velocity of the pulley? Does a similar result hold for the respective linear and angular accelerations?
e) Use the results of parts (d) and (b) to solve for the acceleration of the mass m.
3. (a) Obtain a pulley from the front of the room. The total mass M of the pulley is marked on its side. By measuring the dimensions of the discs that make up the pulley determine its moment of inertia. (Hint: Find the fraction of the total volume that each disc represents and from this determine its mass as a fraction of the total mass of the pulley).
b) Attach the pulley to a clamp and consider a mass m = 20g attached to a cord descending to the groundas shown in the figure. Predict the acceleration of the mass m. Make your prediction when the string is wrapped around the medium ring, and when it is wrapped around the small ring.
c) Test your prediction by allowing the mass m to descend a given distance d and recording the time it takes to reach the ground. Compare with your predicted value. Can you account for any discrepancies?
d) Without the cord attached to the pulley, spin the pulley and record the time it takes to come to rest and how many revolutions it makes during this time. (You need not spin the pulley very rapidly).
e) From your data of part (d) estimate the angular acceleration (due to frictional forces) of the pulley and from this determine the resistive torque that acts on the pulley.
f) Incorporate your result of part (e) into your prediction of part (b) and determine to what extent this improves the agreement with your observed acceleration of the falling mass.