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sunquick
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Homework Statement
A solid copper cylinder, 50 mm long and of 10 mm radius, is suspended in a vacuum calorimeter. Wound on the cylinder is a length of fine copper wire which is used as heater and resistance thermometer. Initially the resistance of the heater is 100.2 Ω. A current of 100 mA is then passed for 5 min, and then, when conditions are steady, the resistance of the heater is found to be 102.5 Ω
Temperature coefficient of resistance of copper = [itex] a_1 = 4.1 \times 10^{-3} K^{-1}[/itex]
Density of copper = [itex] \rho = 8.93 Mg m^{-3}[/itex]
a) What is the specific heat capacity of copper?
b)What assumptions are you making?
c)What is the most important factor limiting the accuracy of the experiment?
d)Why was it necessary to wait for conditions to become steady?
Homework Equations
[tex]Q = c \rho \pi r^2 l \Delta T[/tex]
[tex]P = I R^2 = \frac{dQ}{ dt} [/tex]
[tex]R_1=R_0 (1 + a_1 T)[/tex]
The Attempt at a Solution
For part a:
[tex]R_1=R_0 (1 + a_1 T) <=> \left( \frac{R_1}{R_0} - 1 \right) \frac{1}{a_1} = T = 5.599 K[/tex]
[tex]P = I R^2 = \frac{dQ}{ dt} = (R_0 (1 + a_1 T))^2 I = c \rho l \pi r^2 \frac{dT}{dt}≈ c \rho \pi r^2 l \frac{T}{\Delta t}[/tex]
So I know [itex] \Delta T = 5.599 K [/itex] and
[itex] \Delta t = 5 min = 300 s[/itex]
The problem is that I don't know how either the resistance or temperature vary with time.
I tried to calculate using the average value of resistance
[tex]\bar{R} = \frac{R_0 + R_1}{2} = 101.35 Ω[/tex]
So
[tex]P = I \left(\frac{R_0 + R_1}{2}\right)^2 = 1027,18 W[/tex]
[tex] c = \frac{P}{\rho \pi r^2 l} \left(\frac{T}{\Delta t}\right)^{-1} = 392.6 J K^-1 kg^-1[/tex]
The answer at the back of the book is c = 390 J K^-1 kg^-1 which a bit different from what I got.
Am I missing something or is it just an error due to rounding off?
For part b: I'm assuming that temperature varies linearly with time, and that resistance varies linearly with the temperature.Also I'm assuming that the heat transferred varies linearly with the temperature difference.
For part c: I'd say the most important factor is not knowing exactly how temperature or resistance vary with time, hence having to use average values.
For part d: I don't know why it has the conditions have to become steady. As it anything to do with the fact that once conditions become steady the resistance has a steady value and the transfer of energy rate is constant?
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