Question

In figure show $P_1$ and $P_2$ are massless pulleys $P_1$ is fixed and $P_2$ can move masses of $A,B$ and $C$ are $\frac{9m}{64},2m$ and $m$ respectively. All contains are smooth and its string is massless, $\theta ={\tan }^{-1}\left(\frac{3}{4}\right)$. Find the acceleration of block $C$ in $m/s^2$

Answer 1

Given,

$\tan \theta =\frac{3}{4}$

Tension in the string is $T$

Displacement of C $=y$ (upward)

Displacement of B$=2y$ (downward)

Displacement of A$=\frac{2y}{\tan \theta }=\frac{8y}{3}$

Relation of both acceleration, $a_A=\frac{8a_c}{3}$ (1)

On mass C, $2T-mg=m{a}_c\dots \dots \left(2\right)$

On mass B, $2mg-T=2m{a}_B........\left(3\right)$

On mass A, $m_A{a}_A=m_B{a}_B\tan \theta$

$\Rightarrow a_A=\frac{2m{a}_B\times 3/4}{9m/64}=\frac{32a_B}{3}.........\left(4\right)$

From equation (2) , (3) and (4)

$a_A=\frac{8}{3}(5g-a_c)......\left(5\right)$

Putting equation (5) in equation (1)

$\frac{8}{3}(5g-a_c)=\frac{8a_c}{3}$

$\Rightarrow a_c=\frac{5g}{2}$

Hence, acceleration of $C$ is $\frac{5g}{2}$ upward.