Question

Problem 11

Answer 1

Given information

There are two spheres that have the same mass.

Step-by-step explanation

Step 1.

Consider the spheres roll down an plane that is inclined at an angle with ground without slipping.

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

 

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

The angular velocity of the sphere is

 

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

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

The total energy of the sphere 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 sphere.