Question

From the given figure if speed of end B of rod of length 4 m at an instant when $\theta ={30}^{\circ }$ is 4 m/s then angular velocity of rod is. [angle between wall is ${135}^{\circ }$] Write upto two digits after the decimal point. (Take $\cos 15=\frac{\sqrt{3}+1}{2\sqrt{2}}$ and $\sqrt{3}=1.7$)

Answer 1

Rate of change of $\alpha$ with respect to time is $\omega$.
By Lami's law

$\frac{sin{135}^{\circ }}{4}=\frac{sin\alpha }{x}$

$sin\alpha =\frac{x}{4\sqrt{2}}$ $[\frac{dx}{dt}=4m/s]$

$cos\alpha \frac{d\alpha }{dt}=\frac{4}{4\sqrt{2}}$ $[\frac{d\alpha }{dt}=\omega ]$
and $\alpha ={15}^{\circ }$ when $\theta ={30}^{\circ }$

$cos15\times \omega =\frac{1}{\sqrt{2}}\Rightarrow \omega =\frac{1}{\sqrt{2}}\left\{\frac{2\sqrt{2}}{\sqrt{3}+1}\right]=\frac{2}{\sqrt{3}+1}$

$\omega =0.74rad/s$