Question

The string shown in figure is passing over small smooth pulley rigidly attached to trolley $A$. If the speed of trolley is constant and equal to $v_{A}$ towards right, speed and magnitude of acceleration of block $B$ at the instant shown in figure are

• A. $v_{B}=v_{A}, a_{B}=0$
• B. $a_{B}=0$
• C. $v_{B}=\frac{3}{5} v_{A}$
• D. $a_{B}=\frac{16 v_{A}^{2}}{125}$

Answer 1

Answer: (C), (D)

$(y-h)+\sqrt{x^{2}+h^{2}}=l$
or $\frac{d y}{d t}+\frac{x}{\sqrt{x^{2}+h^{2}}} \frac{d x}{d t}=0$
$\frac{d y}{d t}=-\frac{x}{\sqrt{x^{2}+h^{2}}} \frac{d x}{d t}$
$\Rightarrow \frac{d y}{d t}=-\frac{3}{5}\left(-v_{A}\right)$
$v_{B}=\frac{3}{5} v_{A}$ ...(i)
$\frac{d^{2} y}{d t^{2}}=\frac{v_{A}^{2} h^{2}}{\left(x^{2}+h^{2}\right)^{3 / 2}}$
$\Rightarrow a_{B}=v_{A}^{2} \frac{16}{(5)^{3}}=\frac{16}{125} v_{A}^{2}$ ....(ii)