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A light string fixed at one end to a wooden clamp on a ground passes over a fixed pulley and hangs at the other side. It makes an angle $30^{\circ}$ with the ground. A boy weighting $60 kg$ climbs up the rope. The wooden clamp can tolerate up to vertical force 360 N. Find the maximum acceleration in the upward direction with which the boy can climb safely. The friction of pulley and wooden clamp may be ignored. $(g = 10 m / s^{2})$
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Detailed Solution

Free body diagram

Let T be the tension in rope

From diagram we know

$T - m g = m a - - - - - 1$

${(T \sin θ)}_{\max} = 360$

${(T \sin 30)}_{\max} = 360$

$T_{\max} = 360 \times 2 = 720 N - - - - 2$

Substitute the value of T in equation 1

We get $720 - m g = m a$

$a = \frac{720 - 600}{60} = 2 m / s^{2}$

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