Magnitude of force within thermally expanding restrained bar

[AnnaWerner]

Homework Statement

A steel rod with a length of l = 1.55 m and a cross section of A = 4.52 cm^2 is held fixed at the end points of the rod. What is the size of the force developing inside the steel rod when its temperature is raised by ∆T = 41.0 K? (The coefficient of linear expansion for steel is α = 1.17×10-5 1/K, the Young modulus of steel is E = 200 GPa.)

Homework Equations

F=EA(ΔL/L)
ΔL = L*α*ΔT

The Attempt at a Solution

OK, here is what I have done:
1) Convert GPa to Pa. In my case, 200 GPa = 2*(10^11)Pa. Also, convert A from cm^2 to m^2.
2) Find ΔL: L*α*ΔT (L in m, α, and ΔT in K given).
3) Apply ΔL to F = EA(delta L/L).

Thank you for any assistance.

 

[truesearch]

That looks OK. Put the numbers in and see what you get

 

[AnnaWerner]

Well, I have tried this multiple times, but my answer consistently comes out incorrect. The units are supposed to be Newtons, correct? I'm getting 105,768 N. Seems large . . .

 

[truesearch]

I got ΔL =7.43x10^-4m
Then F = (2x10^11 x 7.43x10^-4 x 4.52x10^-4)/1.55 which came to 43364N
Double check your maths and mine! see if you can spot a difference.

 

[asteriatic]

I would like to provide a more detailed explanation of the solution to this problem.

Firstly, it is important to understand the concept of thermal expansion. When a material is heated, its molecules vibrate more vigorously, causing an increase in the distance between them. This results in an overall increase in the size of the material, known as thermal expansion. The amount of expansion is dependent on the material's coefficient of linear expansion (α), which is a measure of how much the material expands per unit length for every degree of temperature change.

In this problem, we are dealing with a steel rod that is fixed at both ends. This means that it is restrained from expanding freely, and therefore, develops a force within itself as a result of thermal expansion. The magnitude of this force can be calculated using the formula F=EA(ΔL/L), where F is the force, E is the Young's modulus (a measure of the material's stiffness), A is the cross-sectional area of the rod, ΔL is the change in length of the rod due to thermal expansion, and L is the original length of the rod.

To solve this problem, we first need to convert the given values into SI units. The Young's modulus, E, is given in gigapascals (GPa), so we need to convert it to pascals (Pa). This can be done by multiplying E by 10^9. Similarly, the cross-sectional area, A, is given in square centimeters (cm^2), so we need to convert it to square meters (m^2). This can be done by dividing A by 10^4.

Next, we can calculate the change in length, ΔL, using the formula ΔL = L*α*ΔT, where L is the original length of the rod, α is the coefficient of linear expansion for steel, and ΔT is the change in temperature. Substituting the given values, we get ΔL = (1.55 m)*(1.17×10^-5 1/K)*(41.0 K) = 7.52×10^-4 m.

Finally, we can plug in all the values into the formula F=EA(ΔL/L) to calculate the force developed within the steel rod. This gives us F = (2*10^11 Pa)*(4.52*10^-4 m^2)*(7.52*10