Question

The given figure shows a small mass connected to a string, which is attached to a vertical post. If the mass is released from rest when the string is horizontal as shown, the magnitude of the total acceleration of the mass as a function of the angle $\theta$ is

• A. $2g\sin \theta$
• B. $2g\cos \theta$
• C. $g\sqrt{3{\cos }^2\theta +1}$
• D. $g\sqrt{3{\sin }^2\theta +1}$

Answer 1

Answer: (D)

From conservation of energy $\Delta K+\Delta U=0$
$mgh=\frac{1}{2}m{v}^{2}mgl\sin \theta$
$=\frac{1}{2}m{v}^2$
$\Rightarrow a_C=2g\sin \theta =\frac{v^2}{l}=$ radial acceleration
$g\cos \theta =a_t=$ tangential acceleration

Total acceleration
$a=\sqrt{a_c^2+a_t^2}$
$=g\sqrt{{\cos }^2\theta +(2\sin \theta {)}^2}$
$=g\sqrt{\left(1-{\sin }^2\theta \right)+4{\sin }^2\theta }$
$=g\sqrt{1+3{\sin }^2\theta }$