Strain produced in a rod after expansion

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.
  • #1
Hamza Abbasi
47
4

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 !
 
Physics news on Phys.org
  • #2
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?
 
  • #3
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!
 
  • #4
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/
 
  • #5
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 :smile::smile:
 
  • #6
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?
 
  • #7
Hamza Abbasi said:
Why is stress zero?
Because the bar is unconstrained while it is expanding. There are no forces acting on it.
 
  • Like
Likes Hamza Abbasi
  • #8
Thank you for guiding :smile: . Problem solved !
 

What is strain produced in a rod after expansion?

Strain produced in a rod after expansion refers to the change in length or shape of a rod due to external forces, such as heating or stretching. This change is measured in terms of strain, which is the ratio of the change in length to the original length of the rod.

How is strain in a rod after expansion calculated?

Strain in a rod after expansion can be calculated using the formula: strain = change in length / original length. This value is typically expressed as a decimal or percentage.

What factors affect strain produced in a rod after expansion?

The amount of strain produced in a rod after expansion depends on several factors, including the material properties of the rod (such as elasticity and thermal expansion coefficient), the magnitude and direction of the external forces applied, and the temperature at which the expansion occurs.

What are the potential consequences of excessive strain in a rod after expansion?

If a rod experiences excessive strain after expansion, it may result in permanent deformation or even failure. This can cause structural instability and compromise the overall integrity of the rod and any connected components.

How can strain produced in a rod after expansion be minimized or controlled?

To minimize or control strain produced in a rod after expansion, engineers often use materials with lower thermal expansion coefficients, design structures with sufficient flexibility to accommodate expansion, and carefully consider the magnitude and direction of external forces applied to the rod.

Similar threads

  • Introductory Physics Homework Help
Replies
28
Views
1K
  • Introductory Physics Homework Help
Replies
6
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
2K
  • Introductory Physics Homework Help
Replies
14
Views
422
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
11
Views
2K
  • Introductory Physics Homework Help
Replies
11
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
796
  • Introductory Physics Homework Help
Replies
3
Views
922
  • Introductory Physics Homework Help
Replies
13
Views
1K
Back
Top