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.

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Solution

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

Smaller mass, $m_{1} = 8 k g$
Larger mass, $m_{2} = 12 k g$
Tension in the string = T
Mass $m_{2}$ , owing to its weight, moves downward with acceleration a, and mass $m_{1}$ moves
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:
$T - m_{1} g = m_{1} a$ ..........(i)
For mass $m_{2}$ : The equation of motion can be written as:
$m_{2} g - T = m_{2} a$ ........(ii)
Adding equations (i) and (ii), we get:
$(m_{2} - m_{1}) g = (m_{1} + m_{2}) a ∴ a = (\frac{m_{2} - m_{1}}{m_{1} + m_{2}}) g \dots (i i i) = (\frac{12 - 8}{12 + 8}) \times 10 = \frac{4}{20} \times 10 = 2 m / s^{2}$
Therefore, the acceleration of the masses is $2 m / s^{2}$ .
Substituting the value of a in equation (ii), we get:
$m_{2} g - T = m_{2} (\frac{m_{2} - m_{1}}{m_{1} + m_{2}}) g \Rightarrow 12 \times 10 - T = 12 \times 2 \Rightarrow T = 96 N$
Therefore, the tension in the string is 96 N

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