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, m₁ = 8 kg

Larger mass, m₂ = 12 kg

Tension in the string = T

Mass m₂, owing to its weight, moves downward with acceleration a,and mass m₁moves upward.

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

For mass m₁:

The equation of motion can be written as:

T – m₁g = ma … (i)

For mass m₂:

The equation of motion can be written as:

m₂g – T = m₂a … (ii)

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

(m_2-m_1)g = (m_1+ m_2)a

`:.a = ((m_2-m_1)/(m_1+m_2))g ...(iii)`

`=((12-8)/(12+8)) xx 10= 4/20 xx 10 = 2 "m/s"^2`

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

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

`m_2g - T = m_2 ( (m_2-m_1)/(m_1+m_2))g`

`=((2m_1m_2)/(m_1+m_2))g`

`=(2xx12xx8)/(12+8)xx10`

`=(2xx12xx8)/20 xx 10  = 96 N`

Therefore, the tension in the string is 96 N.