In a physical balance working on the principle of moments

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Problem: In a physical balance working on the principle of moments, when 5 mg weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct?  (JEE Mains 2017)

  1. Left arm is longer than the right arm
  2. Both the arms are of same length
  3. Left arm is shorter than the right arm
  4. Every object that is weighed using this balance appears lighter than its actual weight.

Solution: Let $x_l$ and $x_r$ be the lengths of the left arm and the right arm, $m$ be mass of the empty pan and $\Delta m=5$ mg be the additional mass on the left pan to make the beam horizontal.

In a physical balance working on the principle of moment

The weight of the left pan is $(m+\Delta m)g$ and that of the right pan is $mg$. Equating torque of these forces about the suspension point O gives \begin{align} x_l = x_r \left(\frac{m}{m+\Delta m}\right) < x_r,\nonumber \end{align} i.e., left arm is shorter than the right arm.

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