Question

A uniform rod of mass m, length L, area of cross-section A and Young's modulus Y hangs from the ceiling. Its elongation under its own weight will be

• A. zero
• B. $\frac{mgL}{2AY}$
• C. $\frac{mgL}{AY}$
• D. $\frac{2mgL}{AY}$

Answer 1

The correct option is C $\frac{mgL}{AY}$

Weight of the rod $=mg$

Area of cross section of the rod $=A$

Hence

Stress on the rod $=\frac{mg}{A}$

Let the change in length of the rod $=L^{\prime }$

Original length of the rod $=L$

So strain $=\frac{L^{\prime }}{L}$

We know that ,

Youngs modulus $=\frac{stress}{strain}$

$Y=\frac{\left(mg/A\right)}{\left(L^{\prime }/L\right)}$

$Y=\frac{mgL}{A{L}^{\prime }}$

or

$L^{\prime }=\frac{mgL}{AY}$

Hence the change in length under its own weight will be $\frac{mgL}{AY}$