Figure below shows a small mass connected to a string, which is attached to a vertical post. If ball

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5.Work, Energy, Power and Collision

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Figure below shows a small mass connected to a string, which is attached to a vertical post. If the ball is released, when the string is horizontal as shown below. The magnitude of the total acceleration (including radial and tangential) of the mass as a function of the angle $\theta$ is

A

$g \sin \theta$

B

$g \sqrt{3 \cos ^2 \theta+1}$

C

$g \cos \theta$

D

$g \sqrt{3 \sin ^2 \theta+1}$

(KVPY-2011)

Solution

(d)

As ball falls through height $h$ and string turns by angle $\theta$,

Velocity of ball is obtained by equating kinetic and potential energies

$\Rightarrow \frac{1}{2} m v^2 =m g h$

$\Rightarrow v^2=2 g h$

$\Rightarrow v^2=2 g l \sin \theta$

So, radial acceleration is

$a_r=\frac{v^2}{l}=2 g \sin \theta$

'Tangential acceleration is produced by component of weight along the string.

$\therefore$ Tangential acceleration is

$a_t=g \cos \theta$

Hence, total acceleration is

$a=\sqrt{ a _t^2+ a _r^2}$

$=\sqrt{4 g^2 \sin ^2 \theta+g^2 \cos ^2 \theta}$

$=g\sqrt{1+3 \sin ^2 \theta}$

Standard 11

Physics

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