Question

Repeat the preceding problem for a hollow sphere of the same radius and mass and initial speed. Explain the differences in the results.

Reference Problem:
The mass of a hoop of radius 1.0 m is 6.0 kg. It rolls across a horizontal surface with a speed of 10.0 m/s. (a) How much work is required to stop the hoop? (b) If the hoop starts up a surface at 30° to the horizontal with a speed of 10.0 m/s, how far along the incline will it travel before stopping and rolling back down?

(a) More work, less distance
(b) Less work, more distance
(c) More work, more distance
(d) Less work, less distance

Answer 1

A hollow sphere requires less work to stop and travels a shorter distance up an incline than a hoop because it has a lower moment of inertia, resulting in lower rotational kinetic energy for the same mass and radius.

Step-by-step explanation:

When comparing the motion of a hollow sphere and a hoop with the same mass, radius, and initial speed, we must consider the moment of inertia for each object, which reflects how the mass is distributed relative to the axis of rotation. A hollow sphere has a lower moment of inertia compared to a hoop. For these problems involving rolling objects without slipping, the kinetic energy is split into translational and rotational components.

The work required to stop an object in motion across a horizontal surface can be found by considering the total kinetic energy that needs to be dissipated. Since the hollow sphere has a smaller rotational inertia than the hoop, it will require less work to bring it to a stop. When dealing with an incline, the component of gravitational force along the slope decreases the object's kinetic energy, and here too, the hollow sphere's lower moment of inertia means it will travel a shorter distance up the incline before stopping and rolling back down compared to the hoop.

Overall, the hollow sphere requires less work to stop than the hoop and will not travel as far up an incline, due to its smaller moment of inertia.