Question

Problem 12a

Answer 1

Given information

There are one sphere and one cylinder both have the same mass and the same radius.

Step-by-step explanation

Step 1.

Consider both the sphere and cylinder roll down an plane that is inclined at an angle with ground without slipping.

At the top of the plane the object has only potential energy, which is

 

Here, is the total energy at the top, is the mass of the object, is the acceleration due to gravity and is the height.

The angular velocity of the object is

   

Here, is the angular velocity, is the linear speed and is the radius.

At the bottom of the plane, the object has transnational kinetic energy and rotational kinetic energy.

The total energy of the object at the bottom is

   

Here, is the energy and is the moment of inertia.

According to the law of conservation of energy, the total energy of the system remains conserved. Therefore, the energy at the top of the plane is equal to the energy at the bottom.

Equate two formulae of energies and solve to calculate the speed of the object.