A brass rod is 54.2 cm long and an aluminum rod is 64.7 cm long when both rods a

Question

A brass rod is 54.2 cm long and an aluminum rod is 64.7 cm long when both rods are at an initial temperature of 0 degrees Celsius. The rods are placed in line with a gap of 1.1 cm between them. The distance between the far ends of the rods is maintained at 120.0 cm throughout. The temperature is raised until the two rods are barely in contact. The coefficients of linear expansion of brass and aluminum are 2.0*10^-5K^-1 and 2.4*10^-5K^-1, respectively. On figure 14.1a, the temperature at which contact of the rods barely occurs, in degrees Celsius, is closest to: A)417 B)459 C)438 D)375 E)396

Explanation / Answer

The total length of the rods initially is 54.2 + 64.7 = 118.9 cm. The growth must be 1.1 cm (.011 m)

Thus we apply Delta L = Lo(a)Delta T

If we call delta L for brass x, then delta L for aluminum is .011 - x

The delta T's for both will be the same so we can set up the following

x/Lo(a) = (.011 - x)/Lo(a)

x/(.542)(2 X 10^-5) = (.011 -x)/(.647)(2.4 X 10^-5)

x/1.084 X 10^-5 = (.011 - x)/(1.553 X 10^-5)

Cross multiply

1.553 X 10^-5x = 1.1924 X 10^-7 - 1.084 X 10^-5x

x = 4.52 X 10^-3 m

Use that to find the new temp

(4.52 X 10^-3) = (.542)(2 X 10^-5)(T)

T = 417 degrees Celsius

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