Question

Suppose you are the proud owner of a grandfather clock that has a pendulum with a nominal length of 1 meter. You find that the clock runs fast so that it gains 1 second every hour. Should the pendulum be shortened or lengthened to make the clock run on time? By how much?

Answer 1

If it gains 1 second every hour, then it's too fast by 3601 seconds/3600 seconds. As someone pointed out the period of a pendulum clock is T = 2 x pi x sqrt(L/g). Let's define a new period (the correct one) T' = (3601/3600)*T

=> (3601/3600) x 2pi x sqrt(L/g) = 2pi x sqrt(L'/g) where L' is the new length

=> L' = L x (3601/3600)²

Since L is 1m, this means L' - L = .56 mm, which is the distance you need to lengthen it by.