A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking $g = 10 m/s^{2},$ find the work done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest.

Step 1: Identify the given parameters.

The masses of the two blocks and the acceleration due to gravity are given.

$$m_1 = 0.36\ \text{kg}$$ $$m_2 = 0.72\ \text{kg}$$ $$g = 10\ \text{m/s}^2$$

The system is released from rest, so the initial velocity is zero.

$$u = 0$$

Step 2: Write the equations of motion for the blocks.

Since $m_2 > m_1$, block $m_2$ accelerates downwards and block $m_1$ accelerates upwards with the same acceleration $a$. Let $T$ be the tension in the string.

For the block of mass $m_1$:

$$T - m_1 g = m_1 a$$

For the block of mass $m_2$:

$$m_2 g - T = m_2 a$$

Step 3: Calculate the acceleration of the system.

Add the two equations of motion to eliminate $T$.

$$(m_2 - m_1)g = (m_1 + m_2)a$$ $$a = \frac{(m_2 - m_1)g}{m_1 + m_2}$$

Substitute the given values into the expression.

$$a = \frac{(0.72 - 0.36) \times 10}{0.36 + 0.72}$$ $$a = \frac{10}{3}\ \text{m/s}^2$$

Step 4: Calculate the tension in the string.

Express tension $T$ using the equation of motion for $m_1$.

$$T = m_1(g + a)$$

Substitute the values of $m_1$, $g$, and $a$.

$$T = 0.36 \left(10 + \frac{10}{3}\right)$$ $$T = 4.8\ \text{N}$$

Step 5: Calculate the displacement of the $0.36\ \text{kg}$ block.

Use the kinematic equation for displacement during the first second ($t = 1\ \text{s}$).

$$s = \frac{1}{2}at^2$$ $$s = \frac{1}{2} \times \frac{10}{3} \times (1)^2$$ $$s = \frac{5}{3}\ \text{m}$$

Step 6: Calculate the work done by the tension on the $0.36\ \text{kg}$ block.

Since the tension force and the displacement are both in the upward direction, the work done is positive.

$$W = T \cdot s$$ $$W = 4.8 \times \frac{5}{3}$$ $$W = 8\ \text{J}$$

Answer: The work done by the string on the block of mass $0.36\,\text{kg}$ during the first second is $8\ \text{J}$.