Question

A uniform rod of length $2L$, area of cross section $A$ and Young's modulus $Y$ is lying at rest under the action of three forces as shown. Choose the correct statement.

• A. The stress at any cross section is $\frac{2F}{A}$.
• B. The stress at any cross section in section BC is $\frac{F}{A}$.
• C. The total extension is $\frac{3FL}{AY}$.
• D. The total elongation is $\frac{6FL}{AY}$.

Answer 1

The correct option is C The total extension is $\frac{3FL}{AY}$.

As sum of net external force is $2F+\left(-F\right)+\left(-F\right)=0.$Therefore the body is at rest and across any cross-section, net force will be zero.
By cutting cross section along B and drawing FBD it can be established that 2F force tensile in nature acts on both ends at section BC and F force tensile in nature acts in both direction at section AB.
Therefore stress in section AB is $\frac{F}{A}$ and extension in AB is $\frac{FL}{AY}$.
The stress in section BC is $\frac{2F}{A}$ and extension is given by $\frac{2FL}{AY}$.
Net extension can be algebraically added .So, total elongation will be $\frac{FL}{AY}+\frac{2FL}{AY}=\frac{3FL}{AY}.$