Question

A stone tied to an inextensible string of length $l=1\text{m}$ is kept horizontal. If it is released, find the angular speed (rad/s) of the stone when the string makes an angle $\theta ={30}^{\circ }$ with horizontal.

• A. $3.16$
• B. $3$
• C. $2.28$
• D. $4.16$

Answer 1

The correct option is A $3.16$

From Diagram
$a_t=g\cos \theta$
$\alpha \cdot 1=g\cos \theta$
$\alpha =g\cos \theta$
$\frac{\omega d\omega }{d\theta }=g\cos \theta$
${\int }_0^{\omega }\omega d\omega ={\int }_{0^0}^{{30}^0}g\cos \theta d\theta$
$\frac{{\omega }^2}{2}=[g\sin \theta {]}_0^{30}$
$\frac{{\omega }^2}{2}=g[\frac{1}{2}-0]$
$\frac{{\omega }^2}{2}=g\left[\frac{1}{2}\right]$
$\begin{array}{cc}{\omega }^2& =g\\ \omega & =\sqrt{g}\\ & =\sqrt{10}\end{array}$