Question

A particle originally at rest at the highest point of $a$ smooth vertical circle is slightly displaced. It will leave the circle at vertical distance $h$ below the highest point. such that

• A. $h=R$
• B. $h=\frac{R}{3}$
• C. $h=\frac{R}{2}$
• D. $h=2R$

Answer 1

Answer: (B)

$h=\frac{R}{3}$

Ans B

Let at $T$ it leaves the circle

Mechanical energy $=C$

$\begin{array}{cc}{P}_i+k_i& =P_f+K_f\\ mgh& =\frac{1}{2}m{v}^2\end{array}$

$2mgh=m{v}^2---\left(1\right)$

Net force towards 0 is $=\frac{m{v}^2}{R}$

$\mathrm{mgcos}\theta =\frac{m{v}^2}{R}$

$\begin{array}{cc}\mathrm{mg}\left(\frac{R-h}{R}\right)& =2mgh/R\\ R-h& =2h\\ R/3& =h\end{array}$