Question

Attempt 4 A uniform, solid sphere of radius 5.50 cm and mass 1.25 kg starts with a purely translational speed of 3.25 m/s at the top of an inclined plane. The surface of the incline is 2.50 m long, and is tilted at an angle of 34.0∘ with respect to the horizontal. Assuming the sphere rolls without slipping down the incline, calculate the sphere's final translational speed ?2 at the bottom of the ramp.

Answer 1

Let the mass of the sphere be m and its radius r. Suppose the linear speed of the sphere when it reaches the bottom is v. As the sphere rolls down without slipping, its angular speed about its axis is .

Then the Kinetic energy of a body in combined rotation and translation (K) at te bottom will be is the sum of kinetic energy due to rotation and kinetic energy due to translation.

Kinetic energy due to rotation =

Kinetic energy due to translation =

Therefore ----------------- (1)

The moment of inertia of a solid sphere about its diameter (I) is given by, I =

Therefore (1) becomes,

  

This should be equal to the loss of the potential energy,

where g is the acceleration due to gravity = 9.8

Here length l = 2.5 m and angle of inclination =34°

Therefore the final translational speed,

  

  = 4.42 m/s

(Mass and radius of the sphere is given in the question in order to find the moment of inertia of the sphere separately. ie, = )