An inextensible massless string goes over a frictionless pulley. Two weights of 100 N and 200 N are attached to the two ends of the string. The weights are released from rest, and start moving due to gravity. The tension in the string (in N) is

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Solution

Method I:

Tension in string
$[\frac{2 m_{1} m_{2}}{m_{1} + m_{2}}] g$
$⎡ ⎢ ⎢ ⎢ ⎣ \frac{2 \times \frac{100}{g} \times \frac{200}{g}}{\frac{100}{g} + \frac{200}{g}} ⎤ ⎥ ⎥ ⎥ ⎦ g$
$⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{\frac{2 \times 100 \times 200}{g^{2}}}{\frac{(100 + 200)}{g}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ \times g$
$\frac{2 \times 100 \times 200}{300} = \frac{400}{3} = 133.33 N$
Method II:

Assume weight starts moving with acceleration 'a', while tension in string is $T_{1}$
$∴$ Free body diagram of block having 10 N weight,

$T_{1} - 100 = \frac{100 a}{g} . . . (i)$
$∴$ Free body diagram of block having 200 N

$200 - T_{1} = \frac{200 a}{g} . . . (i i)$
Dividing equation (i) with equation (ii)
$\frac{T_{1} - 100}{200 - T_{1}} = \frac{100}{200}$
$2 T_{1} - 200 = 200 - T_{1}$
$3 T_{1} = 400$
Tension in string,
$T_{1} = \frac{400}{3} = 133.33 N$

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