# Strain produced in a rod after expansion

• Hamza Abbasi
In summary, a rod of length ##L_o## with a coefficient of linear expansion of ##\alpha## is placed on a friction-less surface. When the temperature of the rod is increased by ##\Delta T##, the strain developed in the rod is given by the equation $$Strain= \alpha \Delta T$$. The stress developed in the rod will be zero due to the absence of any external forces acting on the unconstrained rod.

## Homework Statement

A rod of length ##L_o## is kept on a friction-less surface. The coefficient of linear expansion for the material of the rod is ##\alpha##. The the temperature of the rod is increased by ##\Delta T## the strain developed in the rod will be?

## Homework Equations

1. $$\Delta L= L_o(1+\alpha \Delta T)$$
2. $$Strain (Linear ) = \frac{\Delta L}{ L_o}$$

## The Attempt at a Solution

$$Strain= \frac{ L_o(1+\alpha \Delta T)}{L_o}$$
$$Strain =(1+\alpha \Delta T)$$

Whereas the answer in book is zero !

Your answer is incorrect, and so is the answer in the book. The first equation should read $$\Delta L=L_0(1+\alpha \Delta T-L_0=L_0\alpha \Delta T$$So the strain is just ##\alpha \Delta T##. Are you sure they weren't asking for the stress?

The book answer is wrong. The stress developed in the rod will be zero if the surface is frictionless, but the strain won't.
Your answer is also wrong. Equation 1 should be L = L0(1 + αΔT). Then ΔL = L - L0.
(Note this is an approximation that applies when ΔT is small. What if it is large?)

Edit: Beat me to it!

Chestermiller said:
Your answer is incorrect, and so is the answer in the book. The first equation should read $$\Delta L=L_0(1+\alpha \Delta T-L_0=L_0\alpha \Delta T$$So the strain is just ##\alpha \Delta T##. Are you sure they weren't asking for the stress?
Yes , I am sure . Question was about strain/

mjc123 said:
The book answer is wrong. The stress developed in the rod will be zero if the surface is frictionless, but the strain won't.
Your answer is also wrong. Equation 1 should be L = L0(1 + αΔT). Then ΔL = L - L0.
(Note this is an approximation that applies when ΔT is small. What if it is large?)

Edit: Beat me to it!
Oh yes! I wrote equation 1 wrong !
Got it  mjc123 said:
The book answer is wrong. The stress developed in the rod will be zero if the surface is frictionless, but the strain won't.
Your answer is also wrong. Equation 1 should be L = L0(1 + αΔT). Then ΔL = L - L0.
(Note this is an approximation that applies when ΔT is small. What if it is large?)

Edit: Beat me to it!
Why is stress zero?

Hamza Abbasi said:
Why is stress zero?
Because the bar is unconstrained while it is expanding. There are no forces acting on it.

• Hamza Abbasi
Thank you for guiding . Problem solved !