Question

A block slides down a smooth inclined plane to the ground when released at the top, in time $tsec$. Another block is dropped vertically from the same point in the absence of the inclined plane and reaches the ground in $t/2sec$. Then the angle of the inclination of the plane with the vertical is:

• A. ${30}^o$
• B. ${45}^o$
• C. ${60}^o$
• D. ${75}^o$

Answer 1

The correct option is A ${30}^o$

Let height be $h$ and the slope be $\theta$
$\Rightarrow l=\frac{h}{\sin \theta }$
Component of $g$ along the inclined $=g\sin \theta$
$\therefore l=\frac{1}{2}g\sin \theta \cdot t^2$
$\frac{h}{\sin \theta }=\frac{1}{2}g\sin \theta \cdot t^2$
$\Rightarrow t=\sqrt{\frac{2h}{g}}\times \frac{1}{\sin \theta }$
For second block:
$h=\frac{1}{2}g{t}^{\prime 2}$
$t^{\prime }=\sqrt{\frac{2h}{g}}$
$t^{\prime }=\frac{t}{2}$
Therefore, on simplifying ,
$2=\frac{1}{\sin \theta }$
$\sin \theta =\frac{1}{2}$
$\Rightarrow \theta =\frac{\pi }{6}$