Question

Block B moves to the right with a constant velocity $v_o$ .
the velocity of body A relative to B is

• A. $\frac{v_o}{2}$ ,towards left
• B. $\frac{v_o}{2}$,towards right
• C. $\frac{3v_o}{2}$ ,towards left
• D. $\frac{3v_o}{2}$ ,towards right

Answer 1

The correct option is B $\frac{v_o}{2}$,towards right

Length of the rope $L$ is constant

So,

$L=X_b-X-f+2(X_b+X_a)$

Here $X_a$ is the distance of block $A$ pulley from origin.

Here $X_b$ is the distance of block $B$ pulley from origin.

Here $X_f$ is the distance from the centre.

$L=(X_b-X_f)+2(X_b+X_a)$

$L=3X_b-X_f-2X_a$ ..............(i)

Differentiating equation (i) with respect to $t$,

$\frac{dL}{dt}=\frac{3d{X}_b}{dt}-\frac{d{X}_f}{dt}-\frac{2d{X}_a}{dt}$

$0=3V_b-0-2V_a$

$V_a=\frac{3}{2}{V}_b$

$V_b=V_0$

$V_a=\frac{3}{2}{V}_0$

$V_{ab}=V_a-V_b$

$=\frac{3}{2}{V}_b-V_b$

$=\frac{V_b}{2}=\frac{V_0}{2}$ Towards right as $V_{ab}$ is positive.