Question

A cube of mass starts at rest from point at a height , where radius of the circular track. The cube slides down the frictionless track and around the loop. The force which the track exerts on the cube at point is:

• A. $3mg$
• B. $mg$
• C. $2mg$
• D. $cubewillnotreachthepoint2$

Answer 1

The correct option is A $3mg$

Applying Conservation of Energy:

Total mechanical energy at $1=$ Total mechanical energy at $2$

$\Rightarrow K_1+P_1=K_2+P_2$

$\Rightarrow 0+mg\left(4R\right)=\frac{1}{2}m{v}^2+mg\left(2R\right)$ (Taking ground as reference)

$\Rightarrow \frac{1}{2}{v}^2=2gR$

$\Rightarrow v=\sqrt{4gR}\to \left(1\right)$

Now forces acting on block at point $2$ from it's own frame are:

$1.$ Normal$\left(N\right)$ due to track downward

$2.$ weight$\left(mg\right)$ downward

$3.$ Centrifugal force $\left(\frac{m{v}^2}{R}\right)$ upward

For vertical equilibrium:

$N+mg=\frac{m{v}^2}{R}$

$\Rightarrow N=\frac{m4gR}{R}-mg$ (Using equation $1$)

$\Rightarrow N=4mg-mg=3mg$

So force exerted by track on block at point $2$ is $3mg$