A chain (mass $M$) is hanging from a wooden structure as shown in the figure $\mathrm{A}$. It is slowly taken to a state as shown in the figure $\mathrm{B}$. Calculate the work done by gravity in the process (assume that after reaching the ground, the chain does not form a heap)

• A

  $\frac{\text{Mg}}{4\text{a}+\pi \text{b}}\left[4{\text{a}}^2+{\text{b}}^2+\text{a}\text{b}\pi \right]$
• B

  $\frac{\text{Mg}\text{b}}{4\text{a}+\pi \text{b}}\left[\text{b}-\text{a}\pi \right]$
• C

  $\frac{2\text{Mg}}{2\text{a}+\pi \text{b}}\left[4{\text{a}}^2-{\text{b}}^2+\text{ab}\pi \right]$
• Correct answer
  D

  $\frac{\text{Mg}\text{b}}{4\text{a}+\pi \text{b}}\left[\text{b}+\text{a}\pi \right]$

Solution

Work done by gravity = – (Change in P.E.)

⇒ u_i = u_linear + u_curved

$=\text{Mass per unit length}=\frac{\text{M}}{\text{a}+\frac{\pi \text{b}}{4}}$

${\text{W}}_{\text{gravity}}={\text{u}}_{\text{i}}-{\text{u}}_{\text{f}}=\frac{\text{Mgb}}{4\text{a}+\pi \text{b}}\left[\text{b}+\text{a}\pi \right]$

Hence, the correct answer is option: 'D'.