Question

In the figure shown, blocks $A$ and $B$ move with velocities $v_1$ and $v_2$ along horizontal direction. Find the ratio $v_1/v_2$.

Answer 1

Given,

Initial length of string $=PR+RQ$

From the drawing if change in ${\theta }_1$ and ${\theta }_2$ is very small,

$TR=T{P}^{\prime }\&RQ=RO$

Block A moves $x_1$ & Block B moves $x_2$

Final Length of string $=P^{{}^{\prime }}R+R{Q}^{\prime }=(PR-x_1\cos {\theta }_1)+(RQ+x_2\cos {\theta }_2)$

For instantaneous velocities, change in angle is negligibly small.

Length of string is unstretchable,

So, Initial length = final length

$PR+RQ=(PR-x_1\cos {\theta }_1)+(RQ+x_2\cos {\theta }_2)$

$x_1\cos {\theta }_1=x_2\cos {\theta }_2$

Differentiate with respect to time.

$V_1\cos {\theta }_1=V_2\cos {\theta }_2$

$\frac{V_1}{V_2}=\frac{\cos {\theta }_2}{\cos {\theta }_1}$

Ratio of velocity $\frac{V_1}{V_2}=\frac{\cos {\theta }_2}{\cos {\theta }_1}$