Question

Figure (10-Q6) shows a smooth inclined plane fixed in a car accelerating on a horizontal road. The angle of incline θ is related to the acceleration a of the car as a = g tanθ. If the sphere is set in pure rolling on the incline,
(a) it will continue pure rolling
(b) it will slip down the plane
(c) its linear velocity will increase
(d) its linear velocity will slowly decrease.
Figure

Answer 1

If the sphere is set in pure rolling on the incline, it will continue pure rolling.

Explanation:

From the sphere's free body diagram,

We get ,

Net force along the slope on the sphere,

The net force magnitude acting on an object's mass multiplied by the object's acceleration as shown in the formula below.  

If the net force acts on an object is zero, the object will not accelerate and will be in a state we call it as equilibrium.

                  $\mathrm{F}_{\mathrm{net}}=\mathrm{mg} \sin \theta-\mathrm{macos} \theta$………………eqn ( 1 )

On putting a = g tanθ in equation ( 1 ),  

we get ,      

             $\mathrm{F}_{\mathrm{net}}=0$  

Therefore, option (a) - if the sphere is put on the incline in pure rolling, it'll continue to roll pure is the correct answer.