Question

A block of mass m is tied to string and hung from the ceiling, another block of mass m is vertically beneath At separation l. if both the blocks are given equal and opposite charges q. Find tension and Normal reaction On ground. If they are to be given same and equal charge what should be value of charge so that string becomes slack.

Answer 1

• For equal and opposite charges:

  • Tension: $T = mg + \frac{kq^2}{l^2}$
  • Normal Reaction: $N = mg - \frac{kq^2}{l^2}$

• For same and equal charges (string slack condition):

  • Required charge: $q = \sqrt{\frac{mgl^2}{k}}$

Answer 2

Solution Overview

1. Case 1: Equal and Opposite Charges (say, top = +q, bottom = –q)
   The Coulomb force (attractive) between the charges is

   $$F = \frac{kq^2}{l^2}.$$

   • Top Block:
     Forces acting:

     • Weight $mg$ downward
     • Coulomb force $F$ downward
     • Tension $T$ upward.

     Equilibrium condition:

     $$T = mg + \frac{kq^2}{l^2}.$$

   • Bottom Block (on ground):
     Forces acting:

     • Weight $mg$ downward
     • Coulomb force $F$ upward (since attraction acts along the line joining the charges, pulling the bottom block upward)
     • Normal reaction $N$ upward from the ground.

     Equilibrium condition:

     $$N + \frac{kq^2}{l^2} = mg \quad \Longrightarrow \quad N = mg - \frac{kq^2}{l^2}.$$

2. Case 2: Same and Equal Charges (say, both = +q)
   Now the Coulomb force is repulsive. Its magnitude is still

   $$F = \frac{kq^2}{l^2}.$$

   • Top Block:
     Forces acting:

     • Weight $mg$ downward
     • Coulomb force $F$ upward (repulsion from the bottom block)
     • Tension $T$ upward.

     The equilibrium condition becomes:

     $$T = mg - \frac{kq^2}{l^2}.$$

     For the string to become slack, $T = 0$. Thus,

     $$mg = \frac{kq^2}{l^2} \quad \Longrightarrow \quad q = \sqrt{\frac{mgl^2}{k}}.$$

   (Note: The bottom block will then experience additional downward force due to repulsion, but the problem only asks for the charge required to slack the string.)

Minimal Core Explanation:

1. For opposite charges, top block equilibrium: $T = mg + \frac{kq^2}{l^2}$ and bottom block equilibrium: $N = mg - \frac{kq^2}{l^2}$.
2. For same charges, top block equilibrium: $T = mg - \frac{kq^2}{l^2}$. String becomes slack when $T=0$, hence $mg = \frac{kq^2}{l^2}$ which gives $q = \sqrt{\frac{mgl^2}{k}}$.