A uniform chain of length \(2 \mathrm{~m}\) is kept on a table such that a length of \(50 \mathrm{~cm}\) hangs freely from the edge of the table. The total mass of the chain is \(5 \mathrm{~kg}\). What is the work done in pulling the entire chain on the table. $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{2}\right)$ (A) \(7.2 \mathrm{~J}\) (B) \(3 \mathrm{~J}\) (C) \(4.6 \mathrm{~J}\) (D) \(120 \mathrm{~J}\)

The length of the chain that is dangling from the table is given as 50 cm. We know that 1m = 100cm, so we convert 50 cm to meters, which gives us 0.5 meters.

This can be calculated by dividing the total mass by the total length of the chain. The total mass is 5 kg, and the total length is 2 m. So, \[ \text{Mass per unit length} = \frac{5 kg}{2 m} = 2.5 \, \frac{kg}{m} \]

This is equal to the mass per unit length multiplied by the length of the chain dangling from the table, and further multiplied by g (acceleration due to gravity), assuming downward direction as negative. So, we get \[ \text{Weight} = -2.5 \frac{kg}{m} \times 0.5m \times 10 \frac{m}{s^2} = -12.5 \, \text{N} \]

Now we find the work done by integrating the varying force (weight) with respect to the distance it acts on. We have to integrate from 0 to 0.5 (length of the dangling part) because the force decreases from -12.5 N (at start) to 0 N (when fully pulled up). The force function in this case is directly proportional to the length of the chain hanging from the table. The integral can be calculated as follow: \[ \text{Work Done} = \int_0^{0.5} (-12.5) \frac{y}{0.5} \, dy = -12.5 \int_0^{0.5} 2y \, dy \] This integration gives -12.5 * [y^2] from 0 to 0.5. Substituting these limits, the resulting work done is \[ \text{Work Done} = -12.5*[ (0.5)^2 - (0)^2] = -12.5 * 0.25 = -3.125 \, \text{J} \] However, work done is a scalar quantity and shouldn't be negative. The negative sign here just indicates that the work was done against the weight of the chain. So, the magnitude of work done will be: \[ \text{Work Done} = 3.125 \, \text{J} \] So, the closest answer is (B) \(3 \mathrm{~J}\).