Q7.18     A solid sphere rolls down two different inclined planes of the same heights but
              different angles of inclination.

             (a) Will it reach the bottom with the same speed in each case? 


Answers (1)

Let the height of the plane is h and mass of the sphere is m.

Let the velocity at the bottom point of incline be v.

So the total energy is given by : 

                             T\ =\ \frac{1}{2}mv^2\ +\ \frac{1}{2}Iw^2

Using the law of conservation of energy :

                                   \frac{1}{2}mv^2\ +\ \frac{1}{2}Iw^2\ =\ mgh

For solid sphere moment of inertia is : 

                                                       I\ =\ \frac{2}{5}mr^2


                                             \frac{1}{2}mv^2\ +\ \frac{1}{2}\left ( \frac{2}{5}mr^2 \right )w^2\ =\ mgh

Put  v  = wr   and solve the above equation. 

We obtain :                              \frac{1}{2}v^2\ +\ \frac{1}{5}v^2\ =\ gh

or                                                   v\ =\ \sqrt{\frac{10}{7}gh}

Thus the sphere will reach the bottom at the same speed since it doesn't depend upon the angle of inclination.