Question

How does thermal expansion affect the accuracy of a pendulum clock?

Answer 1

the clock will become slower by a small amount due to expansion

Explanation:

A pendulum clock has the Time Period

`T = 2.pi. sqrt ( l/g)` where l= length of the pendulum and g the acceleration due to gravity.

By thermal expansion the length of the pendulum will increase say by a small amount `delta(l)`

so the new time period `T' = 2.pi. sqrt {( l +delta (l) )/g) }`

so, `(T') / T` = `sqrt{( l + delta (l)) /sqrt ( l)}`

or `((T') -T ) /T` = `{ sqrt (1 + delta (l)) - sqrt(l)} / sqrt (l)`

so the time period of oscillation will increase , thereby the clock will give time as if slow by (T'-T)

In clocks people use seconds pendulum whose time period os 2 secs and length is about 1m and if the length increases to say 1.001 m

then T'- T = 2 s . {sqrt ( 1.001)- 1 }

so the error can be estimated.