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Thermal Expansion of Rod Problem

  • #1

Homework Statement



In a certain experiment, it is necessary to be able to move a small radioactive source at selected, extremely slow speeds. This is accomplished by fastening the source to one end of an aluminum rod and heating the central section of the rod in a controlled way. If the effective heated section of the rod is 4.7 cm long, at what constant rate must the temperature of the rod be changed if the source is to move at a constant speed of 86 nm/s?


Homework Equations



e = aLT, e=elongation, a = coefficient of thermal expansion, L = initial length, T = change in temperature


The Attempt at a Solution



Since we are trying to determine the rate of change of temperature, I differentiated the equation for thermal expansion in terms of time: de/dt = aL(dT/dt)

The change in length with respect to time (de/dt) is equivalent to the velocity at which the rod is expanding (given as 86 x 10^(-9) m/s). We also know the coefficient of thermal expansion for aluminum (24 x 10^(-6) degC^-1) and the original length (.047 m). Solving for the rate of change of temperature (dT/dt), I got an answer of 0.0762411348 degC/s, however, this answer is incorrect. I am unsure as to where I am going wrong (all units seem correct). Any help would be most appreciated. Thanks.
 

Answers and Replies

  • #2
mgb_phys
Science Advisor
Homework Helper
7,774
12
That's the answer I get (with fewer significant figures) - assuming the rod quickly reaches a uniform temperature.

In 1s you want a change of 86nm in 4.7cm, which is 86e-9/0.047 = 1.89E-6
It lengthens by 24E-6 for each deg C so you want a change of 1.89E-6/24E-6 = 0.079deg/s
 

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