Question

A steel ring of radius r and cross-sectional area A is fitted on to a wooden disc of radius R (R > r). If Young's modulus be Y, then the force with which the steel ring is expanded, is

• A. \( AY \frac{R}{r} \)
• B. \( AY \left(\frac{R - r}{r}\right) \)
• C. \( \frac{Y}{A} \left(\frac{R - r}{r}\right) \)
• D. \( \frac{Yr}{AR} \)

Hint:

For circular rings, strain is based on change in radius (or circumference ratio).

Answer 1

The Correct Option is B

Concept: Young’s modulus: \[ Y = \frac{\text{stress}}{\text{strain}} = \frac{F/A}{\Delta L/L} \]

Step 1: Strain in ring.
\[ \text{Strain} = \frac{R - r}{r} \]

Step 2: Stress.
\[ \text{Stress} = Y \cdot \text{strain} = Y \cdot \frac{R - r}{r} \]

Step 3: Force.
\[ F = \text{stress} \times A = AY \cdot \frac{R - r}{r} \]