Question

Problem 1: A disk of mass M and radius R, a hoop of mass 2M and radius R, and a ball of mass M and radius 2R are rolling without slipping. The hoop can be treated as a thin ring and the ball should be modeled as a hollow sphere. Part (a) The objects are rolling on a flat surface with the same linear speed. Which have the same angular speed? Part (b) The objects are rolling on a flat surface with the same angular speed. Which have the same linear speed? Part (c) Which of the objects has the smallest moment of inertia? Part (d) The objects are placed at the top of an incline and released from rest. Assuming that the objects roll without slipping, which one is first reach the bottom of the incline? Choose the best answer. - the hoop

- the ball
- the disk
- they all reach the bottom at the same time

Answer 1

Part (a) The objects with the same linear speed will have the same angular speed. Part (b) The objects with the same angular speed will have different linear speeds. Part (c) The ball has the smallest moment of inertia. Part (d) The hoop will reach the bottom first when released from an incline.

Step-by-step explanation:

Part (a): If the objects are rolling with the same linear speed, then they will have the same angular speed. This is because the angular speed is directly proportional to the linear speed. Since the linear speed is the same for all the objects, their angular speeds will also be the same.

Part (b): If the objects are rolling with the same angular speed, then their linear speeds will be different. The linear speed is inversely proportional to the radius of the object, so the object with the smaller radius will have a greater linear speed.

Part (c): The moment of inertia depends on the mass and distribution of mass of an object. The object with the smallest moment of inertia will be the ball, as it has the smallest radius and most of its mass concentrated at the surface.

Part (d): When the objects are released from rest on an incline, the one that will reach the bottom first is the hoop. This is because the hoop has the smallest moment of inertia and will accelerate faster compared to the other objects.