Question

A block of mass $m$ is projected on a smooth horizontal circular track with speed $v$. What is the average normal force (in Newtons) exerted by circular walls on the block during its motion from $A$ to $B$?

• A. $\frac{2\sqrt{2}m{v}^2}{\pi R}$
• B. $\frac{m{v}^2}{2\pi R}$
• C. $\frac{m{v}^2}{\sqrt{2}\pi R}$
• D. $\frac{m{v}^2}{R}$

Answer 1

The correct option is A $\frac{2\sqrt{2}m{v}^2}{\pi R}$

$\because$ Avg. normal force = Rate of change in momentum during its motion from $A$ to $B$


$F_N=\frac{\Delta \left(mv\right)}{\Delta t}$
Velocity at point $A$ is $v\hat{i}$
Velocity at point $B$ is $v\hat{j}$
Time taken to move $A$ to $B$
$\Delta t=\frac{\frac{\left(2\pi R\right)}{4}}{v}=\frac{\pi R}{2v}$
$F_N=\frac{m\Delta v}{\Delta t}=\frac{m(v\hat{j}-v\hat{i})}{\frac{\left(\pi R\right)}{\left(2v\right)}}$
$|F_N|=\frac{2mv}{\pi R}\times \sqrt{v^2+v^2}=\frac{2\sqrt{2}m{v}^2}{\pi R}$
$|F_N|=\frac{2\sqrt{2}m{v}^2}{\pi R}N$