Question

A sphere with the same mass and radius as the original cylinder, but a smaller rotational inertia, is released from rest from the top of the ramp. KS and KC are the sphere's and cylinder's total kinetic energy at the bottom of the ramp, respectively. How do KS and KC compare, and why

Answer 1

Answer:

KS < KC

Explanation:

Answer 2

Answer:

The Kinetic energy of Sphere is higher than the cylinder.

( KS > KC )

Explanation:

Given - A sphere with the same mass and radius as the original cylinder, but a smaller rotational inertia, is released from rest from the top of the ramp. KS and KC are the sphere's and cylinder's total kinetic energy at the bottom of the ramp, respectively.

To find - How do KS and KC compare, and why ?

Proof -

We know that,

The total energy of an object =  Potential energy +  linear kinetic energy + rotational kinetic energy.

⇒E = mgh + $\frac{1}{2} mv^{2}$ + $\frac{1}{2} l\omega^{2}$  

Now,

Mass of sphere = m

Radius of sphere = r

So,

The moment of inertia of a uniform solid sphere = $\frac{2}{5} mr^{2}$

Also,

Mass of cylinder = m

Radius of cylinder = r

So,

The moment of inertia of a uniform solid cylinder = $\frac{1}{2} mr^{2}$

Now,

Total energy for the sphere , Es = mgh + $\frac{7}{10} mv^{2}$

Total energy for the cylinder, Ec =  mgh + $\frac{3}{4} mv^{2}$

As they always have the same total energy,

So, for height h of the sphere's velocity has to be higher.

Therefore,

The Kinetic energy of Sphere is higher than the cylinder.