Question

A ball rolls down an inclined plane shown in figure. The ball is first released from rest from P and then later from Q. Which of the following statement is/are correct ?
(i) The ball takes twice as much time to roll from Q to O as it does to roll from P to O.
(ii) The acceleration of the ball at Q is twice as large as the acceleration at P.
(iii) The ball has twice as much K.E at O when rolling from Q as it does when rolling from P.

• A. i and ii only
• B. ii and iii only
• C. i only
• D. iii only

Answer 1

The correct option is D iii only

Force acting on the ball is same in both the cases, therefore acceleration will also be same $a_1=a_2$. Thus, (ii) is wrong.
$S=Ut+\frac{a{t}^2}{2}$
Initially ball is at rest, $U=0$
$S=\frac{a{t}^2}{2}$
So, we see that: $S\propto t^2$
Now, as the distance travelled by the two balls are $S_{1}sin\theta =h$ and $S_{2}sin\theta =2h$
$\frac{S_1}{S_2}=\frac{1}{2}$
$S_2=2S_1$
So, we get: $t_2=\sqrt{2}{t}_1$
Thus (i) is wrong.

The acceleration in both cases i.e at position P as well as Q is $g$, acting in vertically downward direction. Thus (ii) is also wrong.

Now by Conservation of Energy formula, the potential energy possessed by the ball is converted to its Kinetic energy at the bottom of the inclined plane.

PE at point Q = mg(2h) ; PE at point P = mg(h)

Thus, KE at O (when ball falls from Q) = KE at O (when ball falls from P)

So, (iii) is correct.