Find how far ahead a grandfather clock gets after a period of time

[Nathan B]

Homework Statement

A pendulum shortens due to a change in temperature, decreasing the length L and therefore period T. How many seconds ahead does the clock get in 24 hours? We assume that the grandfather clock is completely accurate at a normal pendulum length.
L_i = 1.3 m
ΔT = -10°C

Homework Equations

Equation for the period of a pendulum, T = 2π√(L/G)
Equation for change in length due to temperature. ΔL = αL_iΔT

The Attempt at a Solution

The new and old periods are relatively easy to calculate.

I've gotten all sorts of approximate values for time off, but I need to be exact. One of several methods that I've tried:

T₂ / T₁ = time given by the shortened pendulum/actual time

If we want to examine the results of 24 hours of time passing, we take the number of seconds in 24 hours to be our actual time = 84600 seconds.

Time given by shortened pendulum = 84600*T₂ / T₁

subtract actual time and we know how far off we are:

Time off in seconds = 84600*T₂ / T₁ - 84600

This seems like it should work, but it's always a little bit off.

What can I do to make this more accurate?

 

[haruspex]

This seems like it should work, but it's always a little bit off.

What values are you using for g and α, what answer do you get, and what do you think is a more accurate answer?

 

[Nathan B]

Turns out it was correct, I had a mistake elsewhere in the problem. Thanks for the willingness to help!