Question

The ratio of elongation of a prismatic bar due to its own weight and that of a conical bar of the same length is ............

• A. \( \frac{1}{2} \)
• B. 3
• C. \( \frac{1}{3} \)
• D. 2

Hint:

For bars with varying cross-sectional areas, like conical bars, the elongation is not linear and must account for the changing cross-section. The prismatic bar elongates more uniformly.

Answer 1

The Correct Option is B

The elongation \( \Delta L \) of a bar due to its own weight is given by the formula: \[ \Delta L = \frac{W L}{A E} \] where:
- \( W \) is the weight of the bar,
- \( L \) is the length of the bar,
- \( A \) is the cross-sectional area of the bar,
- \( E \) is the Young's modulus of the material.
For a prismatic bar with a constant cross-sectional area \( A \), the weight \( W \) is proportional to \( A \) (i.e., \( W = \rho g A L \), where \( \rho \) is the material density and \( g \) is the acceleration due to gravity).
For a conical bar with a variable cross-section, the area at any point along the length varies. Specifically, the cross-sectional area at the bottom of the bar is larger than at the top. The elongation of a conical bar can be calculated using an integrated approach considering the varying cross-section.
Upon comparing the two elongations for a prismatic and a conical bar of the same length and material, the elongation of the conical bar will be three times that of the prismatic bar. Hence, the ratio of elongation of the prismatic bar to the conical bar is 3.