Question

Two masses $8$ $kg$ and $12$ $kg$ are connected at the two ends of a light inextensible string that goes over a frictionless pulley. Find the acceleration of the masses, and the tension in the string when the masses are released.

Answer 1

The given system of two masses and a pulley can be represented as shown in the following figure:

Smaller mass, $m_1=8$ kg

Larger mass,$m_2=12$kg

Tension in the string$=T$

Mass $m_2$, owing to its weight, moves downward with acceleration, and mass $m_1$ moves upward.

Applying Newton’s second law of motion to the system of each mass:

For mass $m_1$:

The equation of motion can be written as:

$T–m_{1}g=ma$… (i)

For mass m2:

The equation of motion can be written as:

$m_{2}g–T=m2a$ … (ii)

Adding equations (i) and (ii), we get:

$(m2-m1)g=(m1+m2)a$

$a=\frac{(m2-m1)g}{(m1+m2)}$ ....(iii)

$=\frac{(12-8)}{(12+8)}\times 10=\frac{4\times 10}{20}=2ms-2$

Therefore, the acceleration of the masses is 2 m/s2.

Substituting the value of a in equation (ii), we get:

$m_{2}g-T=m_2\frac{(m_2-m_1)g}{(m_1+m_2)}$

$T=(m2-\frac{(m_2^2-m1m2)}{(m1+m2)}g$

$=\frac{2m_1{m}_{2}g}{(m_1+m_2)}$

$T=2\times 12\times 8\times 10/(12+8)$

$T=96N$

Therefore, the tension in the string is 96 N.