Question

A smooth track of incline of length $l$ is joined smoothly with circular track of radius $R$. A mass of $m$kg is projected up from the bottom of the inclined plane. The minimum speed of the mass to reach the top of the track is given by, $v=$

• A. ${\left[2g\right(l\cos \theta +R\left)\right(1+\cos \theta \left)\right]}^{1/2}$
• B. ${(2gl\sin \theta +R)}^{1/2}$
• C. ${\left[2g\right\{l\sin \theta +R(1-\cos \theta )\left\}\right]}^{1/2}$
• D. ${(2gl\cos \theta +R)}^{1/2}$

Answer 1

The correct option is C ${\left[2g\right\{l\sin \theta +R(1-\cos \theta )\left\}\right]}^{1/2}$

When the mass reach to the top , the final velocity $=0$ and height, $H=h+h^{\prime }$
or, $H=l\sin \theta +R(1-\cos \theta )$
using, $v^2-u^2=2aS$, we get
$0-u^2=2(-g)H\Rightarrow u^2=2g[l\sin \theta +R(1-\cos \theta \left)\right]$
or $u={\left[2g\right\{l\sin \theta +R(1-\cos \theta )\left\}\right]}^{1/2}$