Question

.A bucket with mass m2 and a block with mass m1 are hung on a pulley

system. Find the magnitude of the acceleration with which the bucket and

the block are moving and the magnitude of tension force T by which the

rope is stressed. Ignore the masses of the pulley system and the rope. The

bucket moves up and block moves down.​

Answer 1

Let $\sf{a}$ be the net acceleration of the system. This is the individual net acceleration of the block and bucket each, since the string undergoes no extension or compression.

The block experiences weight $\sf{m_1g}$ downwards and the tension in the string $\sf{T}$ upwards.

Since it moves down, the net force acting on the block is given by,

$\sf{\longrightarrow m_1a=m_1g-T\quad\quad\dots(1)}$

The bucket experiences weight $\sf{m_2g}$ downwards and the tension in the string $\sf{T}$ upwards.

Since it moves up, the net force acting on the bucket is given by,

$\sf{\longrightarrow m_2a=T-m_2g\quad\quad\dots(2)}$

Adding (1) and (2),

$\sf{\longrightarrow m_1a+m_2a=m_1g-T+T-m_2g}$

$\sf{\longrightarrow (m_1+m_2)a=(m_1-m_2)g}$

$\sf{\longrightarrow\underline{\underline{a=\left(\dfrac{m_1-m_2}{m_1+m_2}\right)g}}}$

Putting value of $\sf{a}$ in (1),

$\sf{\longrightarrow m_1\left(\dfrac{m_1-m_2}{m_1+m_2}\right)g=m_1g-T}$

$\sf{\longrightarrow T=m_1\left(1-\dfrac{m_1-m_2}{m_1+m_2}\right)g}$

$\sf{\longrightarrow\underline{\underline{T=\dfrac{2m_1m_2g}{m_1+m_2}}}}$