Question

Considered the situation as shown in the figure. A block is released from height $h$ on a smooth track. Find the minimum value of $h$ so that it completes the vertical circle.

Answer 1

REF.Image.

$T-mg=m{a}_0$

In order to complete a full circle,

the normal face when the block

is at the height point of circular

ramp should be greater than zero

So,

if vel at point B =$V_{B0}$ Then,

At point B,$mg+N-\frac{m{v}_B^2}{R}$

As, $N>0$

$\Rightarrow v_B^2>mgR$

At point -A,

$\frac{mv{A}^2}{2}-\frac{mv{B}^2}{2}=mg\left(2R\right)$ [As $△KE=△PE]$

$\frac{mv{A}^2}{2}=2mgR+\frac{mv{B}^2}{2}$

As $V{B}^2>mgR$

$\Rightarrow \frac{mv{A}^2}{2}>2mgR+\frac{mgR}{2}\Rightarrow \frac{m{v}_A^2}{2}>\frac{5mgR}{2}$

Also, $\frac{MV{a}^2}{2}-0=mgh$

As, $\frac{mv{A}^2}{2}>\frac{5mgR}{2}\Rightarrow mgh>\frac{5mgR}{2}$

$\Rightarrow h>5/2R$