Question

A steel cylinder of diameter 100 mm and a copper tube of an outer diameter 200 mm are compressed between the plates of a press. If the ratio of their moduli (steel to copper) is 15/8, what is the ratio of their stresses in steel (\( \sigma_s \)) and copper (\( \sigma_c \))?

• A. \( \frac{15}{8} \)
• B. \( \frac{8}{15} \)
• C. \( \frac{1}{2} \)
• D. 2

Hint:

When dealing with problems involving deformation, use the relationship between stress, modulus, and elongation to establish the correct ratios.

Answer 1

The Correct Option is A

Let the stresses in steel and copper be \( \sigma_s \) and \( \sigma_c \), respectively. From the condition of equal deformation under the applied load, the elongations in both steel and copper must be the same.
The elongation in a material is given by the formula: \[ \Delta L = \frac{F L}{A E} \] Where: - \( F \) is the force, - \( L \) is the length, - \( A \) is the cross-sectional area, - \( E \) is the modulus of elasticity. The total force \( F \) applied is the same for both materials, so the elongations will be equal. Thus, the ratio of the stresses in steel to copper is the inverse of the ratio of their moduli: \[ \frac{\sigma_s}{\sigma_c} = \frac{E_c}{E_s} = \frac{15}{8} \] Hence, the correct answer is \( \frac{15}{8} \).