Class 11 Physics Mechanical Properties of Solids Longitudinal stress

Types of Stress: Longitudinal stress

• Longitudinal stress is defined as restoring force per unit area when the force is applied to the cross-sectional area of the cylindrical body.
• Consider a cylinder which we have to deform. If we apply the force perpendicular to the cross-sectional area, there will be a restoring force that develops in the cylinder in the opposite direction.
• This restoring force per unit area is known as longitudinal stress.
• Experimentally we can observe the increase in length.
• If we tie a heavy object to the cylinder with the help of threads.
• Let Initial length of the cylinder is L.
• After it gets stretched its length increases by ΔL due to the stress.
• As there is change in the length therefore this type of stress is known as longitudinal stress.
• In the below figure if we attach a box to the cylinder, a force is applied on the cross-sectional area of cylinder due to which it gets stretched and as a result there is change in the length of the cylinder.

Problem:- A 14.5 kg mass, fastened to the end of a steel wire of unstretched length 1.0 m, is whirled in a vertical circle with an angular velocity of 2 rev/s at the bottom of the circle. The cross-sectional area of the wire is 0.065 cm2. Calculate the elongation of the wire when the mass is at the lowest point of its path.

Answer:- Mass, m = 14.5 kg

Length of the steel wire, l = 1.0 m

Angular velocity, ω = 2 rev/s

Cross-sectional area of the wire, a = 0.065 cm2

Let Δl be the elongation of the wire when the mass is at the lowest point of its path.

When the mass is placed at the position of the vertical circle, the total force on the mass is:

F = mg + mlω2

= 14.5 × 9.8 + 14.5 × 1 × (2)2 = 200.1 N

Young’s modulus =Stress/Strain

Y = (F/A)/ Δl/l

= (F l)/A Δl

Therefore Δl = F l/A Y

Young’s modulus for steel = 2 × 1011 Pa

Therefore Δl = 200.1/0.065x10-4x2x1011 = 1539.23x107

=1.539x10-4m

Hence, the elongation of the wire is 1.539 × 10–4 m.

Types of Longitudinal Stress:-

1. Tensile Stress
2. Compressive Stress

Tensile Stress

• Tensile stress is a longitudinal stress when the length of the cylinder increases.

For example:-

• When the force is applied to both sides of the cylinder, the cylinder gets stretched. As a result there will be increase in its length.

Force is applied on both the sides as a result length of cylinder increases

Compressive Stress

• Compressive stress is a longitudinal stress where the force is applied to compress the cylinder.

Compressing the cylinder

.