Question

Figure shows a hemisphere and a supported rod. Hemisphere is moving right with a uniform velocity $v_2$ and the end of rod which is in contact with ground is moving left with a velocity $v_1$. The rate at which the angle $\theta$ is decreasing will be

• A. $\frac{(v_1+v_2){\sin }^2\theta }{R\cos \theta }$
• B. $\frac{(v_1+v_2)\tan \theta }{R\cos \theta }$
• C. $\frac{(v_1+v_2){\cos }^2\theta }{R\sin \theta }$
• D. $\frac{(v_1+v_2)\cot \theta }{R\sin \theta }$

Answer 1

The correct option is A $\frac{(v_1+v_2){\sin }^2\theta }{R\cos \theta }$

From right angled triangle formed by the rod,
$\sin \theta =\frac{R}{x}$
$x=Rcosec\theta$
Differentiating above equation, we get
$\frac{dx}{dt}=-R\text{cosec}\theta \cot \theta \frac{d\theta }{dt}$

$\frac{dx}{dt}\text{can be witten as}{v}_1+v_2$
Hence,
$\frac{d\theta }{dt}=\frac{-(v_1+v_2){\sin }^2\theta }{R\cos \theta }$
($-ve$ sign shows that $\theta$ decreasing with time).