Question

A solid homogeneous cylinder and a solid homogeneous sphere each have the same mass and radius. They both roll without slipping, have the same linear speed and approach the same inclined plane. 

As they roll up the inclined plane, they continue to roll without slipping. Which object will reach the greatest height on the inclined plane?

A: solid homogeneous cylinder
B: solid homogeneous sphere
C: they both reach the same maximum height

If you could explain why in your answer that would be great!

Answer 1

We can analyze the problem using energy conservation
The kinetic energy of the two objects before the climb are the same because the initial velocity is same
KE_linear  = (1/2) m v²
Since the balls are rolling
v = r
Where r is the radius of the object , is the angular frequency of the object
The rotational kinetic energy of the ball
KE_rot  = (1/2) I ²
KE_rot  = (1/2) I (v / r)²
Where I is the moment of inertia of the object
The potential energy at the incline is
PE = KE_linear  + KE_rot
mg h =  (1/2) m v²+ (1/2) I (v / r)²​
Since the object's radius and the initial velocity are the same , the height reached depends only on the moment of inertia of the object . The greater the inertia , the higher the object rises
The moment of inertia of the sphere is (2/5) m r²
The moment of inertia of the cylinder is (1/2) m r²
Thus moment of inertia of the cyliner is greater than the moment of inertia of the sphere, the height reached by the cylinder will be higher
The answer is option A