Question

A vertical metal cylinder of radius 2 cm and length 2 m is fixed at the lower end and a load of 100 kg is put on it. Find (a) the stress (b) the strain and (c) the compression of the cylinder. Young modulus of the metal = 2 × 10¹¹ N m⁻².

 

Answer 1

Given:
Radius of cylinder (r) = 2 cm = \[2 \times {10}^{- 2} \text{ m}\]  

Length of cylinder (L) = 2 m
Mass of the load = 100 kg
Young's modulus of the metal = \[2 \times {10}^{11} \text{ N/ m}^2\]

(a) Stress(ρ) is given by :  \[\frac{F}{A}\]

 Here, F is the force given by mg = \[100 \times 10 = 1000 N\] ( Taking g = 10 m/s²)

   A is the area of cross-section = πr²  = 4π \[\times {10}^{- 4} m^2\]

\[\Rightarrow \text{ Stress } \rho = \frac{\text{ mg }}{A}\]
\[ = \frac{\left( 100 \times 10 \right)}{\left( 4\pi \times {10}^{- 4} \right)}\]
\[ = 7 . 96 \times {10}^5 \text{ N/ m}^2\]

(b) Strain is given by:

\[\text{ Strain } = \frac{\rho}{Y} = \frac{\left( 7 . 96 \times {10}^5 \right)}{\left( 2 \times {10}^{11} \right)}\]

\[ = 4 \times {10}^{- 6}\]

(c) Compression of the cylinder:
                           ΔL = strain × L
                                = 4 × 10⁻⁶ × 2 = 8 × 10⁻⁶ m