The Inclined Plane Race: Sphere vs. Hoop

[Tricks67]

a sphere and a hoop both of same mass (e.g. 'm') roll down an inclined plane without slipping...which will get to the bottom first..will they have equal kinetic energies when they reach the bottom?

 

[rcgldr]

This appears to be homework. If so, you're supposed to show some attempt at solving the problem.

 

[Tricks67]

nope not hw..(i wish our homework wud have had such long deadlines)...some problems given in a book...and i post the ones, i can't approach..

 

[Malor]

nope not hw..(i wish our homework wud have had such long deadlines)...some problems given in a book...and i post the ones, i can't approach..

No... though both objects have the same potential Energy at t = 0s, they have different "moments of inertia" and since they are both rolling they have both kinetic and rotational Energy when they reach the bottom of the inclined plane. The hoop will be slower.

E_rot = 1/2 J ω²

 

[asteriatic]

This is an interesting question that can be solved using principles of physics and mathematics. Both the sphere and the hoop have the same mass, so we can assume that they have the same potential energy at the top of the inclined plane. As they roll down the incline, this potential energy is converted into kinetic energy.

The main factor that will determine which object reaches the bottom first is the moment of inertia. The moment of inertia is a measure of how an object's mass is distributed around its axis of rotation. In the case of the sphere, all of its mass is distributed at the same distance from its axis of rotation, so it has a smaller moment of inertia compared to the hoop, where some of the mass is further from the axis of rotation.

This means that the sphere will have a higher angular velocity compared to the hoop as they roll down the inclined plane. This higher angular velocity will result in the sphere covering a greater distance in the same amount of time, allowing it to reach the bottom first.

In terms of kinetic energy, the sphere will have a greater kinetic energy compared to the hoop when they reach the bottom. This is because kinetic energy is directly proportional to the square of an object's velocity. Since the sphere has a higher velocity, it will have a greater kinetic energy.

In conclusion, the sphere will get to the bottom of the inclined plane first and will also have a greater kinetic energy compared to the hoop. This can be confirmed by conducting an experiment and measuring the time taken for each object to reach the bottom, as well as their velocities and kinetic energies.