How Does Temperature Influence Grandfather Clock Timing?

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How does temperature affect grandfather clocks? A metal rod pendulum is inside of the clock. I have a question regarding this. If an expirement happened where 3 clocks were sent to the same time. ONe was put in a cold room, one in a hot, and one in a room temperature room. What would the cold room grandfather clock and the hot room grandfather clock say if the room temperature clock was 3 o'clcok? PLease help me! EXPLAIN
 
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You know that the period of a pendulum depends on its length. What happens to the length of a pendulum under the conditions you specified?

P.S. I won't even point out that all three rooms, by definition, are at "room temperature!" :)
 
I am guessing that the metal rod expands when the room is hotter and "shrinks" when the room is colder. If the length is a dependant factor, wouldn't the period affect it becuase the rod lengthens and condenses?
 
Am I right with the whole lengthening thing?
 
Yes! You know that a longer pendulum has a longer period than a shorter one so you should now be able to completely answer the question.
 
To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.
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