Problem 7 - University Physics with Modern Physics

Question

Consider a cylinder and a hollow cylinder, rotating about an axis going through their centers of mass. If both objects have the same mass and the same radius, which object will have the larger moment of inertia? a) The moment of inertia will be the same for both objects. b) The solid cylinder will have the larger moment of inertia because its mass is uniformly distributed. c) The hollow cylinder will have the larger moment of inertia because its mass is located away from the axis of rotation.

Step-by-Step Solution

Answer

Explain your answer. Answer: The hollow cylinder will have a larger moment of inertia because its mass is located away from the axis of rotation. The moment of inertia of the solid cylinder is half the moment of inertia of the hollow cylinder.

Step 1: Recall the moment of inertia formulas for solid and hollow cylinders

To compare the moment of inertia of both objects, we need to recall their formulas. The moment of inertia (I) of a solid cylinder is given by: \[I_\text{solid} = \frac{1}{2}MR^2,\] where M is the mass of the cylinder and R is its radius. For a hollow cylinder, the moment of inertia is given by: \[I_\text{hollow} = MR^2.\]

Step 2: Compare the moment of inertia formulas

Now, we can compare the moments of inertia of the solid and hollow cylinders: \[\frac{I_\text{solid}}{I_\text{hollow}} = \frac{\frac{1}{2}MR^2}{MR^2} = \frac{1}{2},\] meaning that the moment of inertia of the solid cylinder is half the moment of inertia of the hollow cylinder.

Step 3: Choose the correct answer

Since the moment of inertia of the hollow cylinder is larger than that of the solid cylinder, the correct answer is (c) The hollow cylinder will have the larger moment of inertia because its mass is located away from the axis of rotation.