Question

Part A

If these two objects roll up the slope without slipping, which reaches the greater maximum height?
If these two objects roll up the slope without slipping, which reaches the greater maximum height?
The cylindrical shell
The solid disk
Both reach the same maximum height
More information about the objects' mass and diameter is needed.

Answer 1

The cylindrical shell, having a greater moment of inertia than a solid disk of the same mass and radius, converts less kinetic energy into rotational energy and more into potential energy, allowing it to reach a greater maximum height on the incline.

Step-by-step explanation:

Comparing Maximum Heights of Rolling Objects

When considering two objects rolling up a slope without slipping, such as a cylindrical shell and a solid disk, the amount of rotational inertia plays a significant role in determining which reaches a greater maximum height. Rotational inertia, or the moment of inertia, depends not only on the mass of the object but also on how the mass is distributed relative to the axis of rotation. A hollow cylinder has a greater moment of inertia than a solid disk of equal mass and radius because more of its mass is distributed farther from the centre. This means that assuming equal initial kinetic energy, the solid disk will convert more of its kinetic energy into rotational energy, leaving less energy to be converted into potential energy as it ascends the slope.

Therefore, the cylindrical shell, with its greater moment of inertia, would convert less of its kinetic energy into rotational energy and more into gravitational potential energy, allowing it to ascend to a greater maximum height on the incline than the solid disk. It's essential to note that the question assumes no energy losses due to factors like air resistance or friction apart from what's necessary to prevent slipping. No additional information about mass and diameter is needed because they are stated to be the same for both objects.

Answer 2

Due to the difference in moment of inertia, a solid disk reaches a greater maximum height than a cylindrical shell when both roll up a slope without slipping. The solid disk has less rotational energy and more of its initial energy can be converted into potential energy.

Step-by-step explanation:

When comparing a cylindrical shell and a solid disk rolling up a slope without slipping, we need to understand the concept of rotational inertia or moment of inertia. These two objects differ in their distribution of mass, which affects their moment of inertia. The moment of inertia is larger for a hollow object at the same mass and radius as it has more mass distributed further from the axis of rotation.

During the motion up the slope, mechanical energy is conserved. For an object rolling without slipping, this energy is divided between translational kinetic energy (Kt) and rotational kinetic energy (Kr). Since the cylindrical shell has a greater moment of inertia compared to the solid disk, more of its initial kinetic energy will be stored as rotational kinetic energy.

This means that at the peak of the motion, where all the kinetic energy has been converted into potential energy, the cylindrical shell will have a lower translational kinetic energy at any given height compared to the solid disk. Therefore, the cylindrical shell will reach a lower height compared to the solid disk, making the solid disk the one that reaches the greater maximum height.