# Homework Help: Thermal expansion of a pendulum

1. Apr 1, 2009

### fluidistic

Hi,
I've done the problem but I'm unsure of my answer. I would be glad if you could check it out.
1. The problem statement, all variables and given/known data
A steal pendulum is considered a clock at a certain temperature.
What is the maximum variation of temperature we can submit to the pendulum if it cannot delay more than one second by day?

2. Relevant equations Coefficient of dilation of steal : $$11\times 10 ^{-6}°C^{-1}$$.

3. The attempt at a solution
First I notice that a period of the pendulum corresponds to a second. Then I calculated the number of seconds in day to be 86400.
Say it delays 1 second in a day and I want to calculate its period. We have that $$86400T=86401T'$$. Replacing $$T$$ with $$2\pi \sqrt {\frac{g}{l}}$$ then I get that $$l'=0.9999768523l$$ where $$l$$ is the length of the pendulum and $$l'$$ the length of the heated pendulum.
Now I want to find the length it cannot overpass : $$l(1-0.9999768523)=0.00002314774628l$$.
Looking at the coefficient of dilation of steal, if I heat the pendulum by 1°C, it will grow $$0.0000011l$$. From it, I just look and see that I can heat the pendulum up to 23°C more than it is.
To my intuition it looks too much. What do you say?

2. Apr 1, 2009

### rl.bhat

LaTeX Code: 0.0000011l
Is this correct?

3. Apr 1, 2009

### fluidistic

According to my assignments yes. However I just checked it up on wikipedia thanks to you. (the page is http://en.wikipedia.org/wiki/Coefficient_of_dilatation) and they give a range of $$33.0$$ ~ $$39.0 \times 10 ^{-6}$$. This reduce the temperature I got by 3. So it's around 7°C which makes more sense to me.
EDIT: Ah no! Sorry, it is what my assignment says, around $$11\times 10 ^{-6}°C^{-1}$$ since we're talking about the coefficient of linear thermal expansion...

4. Apr 1, 2009

### rock.freak667

Does the equation $\delta l=L \alpha T$ not apply here? Well that is what I thought to use first when you found the extension.

5. Apr 9, 2009

### fluidistic

Ah I made a little mistake! The coefficient of dilation of steel is $$11 \times 10 ^{-6}°C^{-1}$$.
Quoting myself :
should be
. With this, the answer becomes 2.3 °C which is the right answer. (I asked the professor).
I'm glad I found my mistake.