Several identical square copper rods of dimensions 1 times 1 times 100 cm are in

Question

Several identical square copper rods of dimensions 1 times 1 times 100 cm are in a room sized oven. Some of the rods are laying on the floor, and some are leaning against the wall, more or less vertical. The temperature of the copper is taken from 20 degree C to 400 degree C. Copper has a coefficient of linear thermal expansion of alpha = 16.6 times 10^-6 K^-1. Assuming alpha stays constant over the entire temperature range, what are the new dimensions of the rods? Given the rods cannot expand into the floor, the center of mass of each rod must move up during this process. Where does the energy to lift the center of mass come from? How does the center of mass shift for a rod laying on the floor compare to the shift for a rod leaning against a wall? What does this suggest about the specific heat of a vertical rod Compared to a horizontal rod? Provide an estimate for any difference.

Explanation / Answer

Given
   square copper rod of dimensions 1 X 1X 100 cm
and the temperatures are initial T1 = 20 0C, final temperature T2 = 400 0C

coefficient of linear expansion is alpha = 16.6*10^-6 k^-1

we know that from the definition of alpha

       alpha = DL /(l1(DT)
  
       DL = alpha*l1(DT)
  
       DL = 16.6*10^-6 *1*380 m = 0.006308 m

       l2-l1 = 0.006308
  

       l2 = 0.006308+l1 = 0.006308+1 = 1.006308 m

as the rod is of length 1m the center of mass is at its center that is at 0.5 m

due to change in lenght the center of mass shits up , to 1.006308/2 = 0.503154 m

both the rods have the same center of mass after the expansion

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