Thermal Expansion Problem: Help Solve Clearance Issue

[kris24tf]

Thermal Expansion Please Help!

I have a Thermal Expansion problem that I am having trouble with...

It goes :

A brass rod has a circular cross section of radius .5 cm. The rod fits into a circular hole in a copper sheet with a clearance of .010 mm completely around it when both it and the sheet are at 20 degrees C. At what temperature will the clearance be zero?

I know I have to use Area of Expansion here, but I am not sure which version to use. I knwo to convert .5 cm to .005m and .010mm to 10m. I also know that delta T equals T-293=20-293=-273K. I do not know how to develop the equation. Any assistance would be appreciated, although I will warn that I do not do well with written explanations as well as numerical ones...

 

[Nate810]

Thank you! The equation for the thermal expansion of an object with a circular cross section is:Δx = α(T−T₀)Rwhere α is the coefficient of linear expansion, T is the temperature of the object, T₀ is the initial temperature, and R is the radius of the circular cross section. Using the given values, we can solve for T:0.010 mm = α(T−293)0.005 mT = 293 + (0.010 mm / (α 0.005 m))T = 293 + (10 x 10^-3 m / (α 0.005 m))T = 293 + (20 x 10^-3 m / α)Therefore, the temperature at which the clearance will be zero is 293 + (20 x 10^-3 m / α), where α is the coefficient of linear expansion for brass and copper.

 

[quicknote]

Thank you for reaching out for assistance with your thermal expansion problem. I understand that you are having difficulty developing the equation and are more comfortable with numerical explanations. I will try my best to provide a clear and concise solution to your problem.

First, let's start by understanding the concept of thermal expansion. Thermal expansion is the tendency of matter to change in volume, length, or area in response to a change in temperature. In your problem, the brass rod and the copper sheet will expand or contract with a change in temperature.

To solve this problem, we will be using the formula for thermal expansion in length, which is given by ΔL = αLΔT, where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature.

Now, let's apply this formula to your problem. We know that the original length of the brass rod is 2r, where r is the radius of the rod. So, the original length of the rod is 2(0.005m) = 0.01m.

Next, we need to find the coefficient of linear expansion for brass and copper. The coefficient of linear expansion for brass is 18.7 x 10^-6 m/m*K and for copper is 16.6 x 10^-6 m/m*K. We will use these values in our equation.

Now, we can solve for the change in length of the rod using the formula ΔL = αLΔT. Plugging in our values, we get:

ΔL = (18.7 x 10^-6 m/m*K)(0.01m)(ΔT)

Since we want to find the temperature at which the clearance will be zero, we can set the change in length to zero and solve for ΔT.

0 = (18.7 x 10^-6 m/m*K)(0.01m)(ΔT)

ΔT = 0 K

This means that at a temperature of 0 K or -273°C, the clearance will be zero. This is known as the absolute zero temperature, where all thermal expansion stops.

In conclusion, using the formula for thermal expansion in length and the coefficients of linear expansion for brass and copper, we were able to solve for the temperature at which the clearance will be zero. I hope this explanation helps and if you have any further questions, please do