First slide

Vertical circular motion

Question

A small block slides with velocity $v_{0} = 0.5 \sqrt{gr}$ on the horizontal frictionless surface as shown in the fig. The block
leaves the surface at point C. The angle $θ$ in the figure is:

Moderate

A

$\cos^{- 1} \frac{4}{9}$
B

$\cos^{- 1} \frac{3}{4}$
C

$\cos^{- 1} \frac{1}{4}$
D

$\cos^{- 1} \frac{4}{5}$

Solution

At point C: $mg \cos θ - N = \frac{{mv}^{2}}{r}$
(when block leaves the surface normal force becomes zero, so putting $N = 0) \Rightarrow g \cos θ = \frac{v^{2}}{r}$
$v^{2} = {v^{2}}_{0} + 2 gh, h = r - r \cos θ$
$gr cos θ = (0.5 \sqrt{gr})^{2} + 2 g (r - r \cos θ)$
$\Rightarrow \cos θ = \frac{1}{4} + 2 - 2 \cos θ \Rightarrow \cos θ = \frac{3}{4}$

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