Question

If you take a pendulum clock from Paris to Cayenne, French Guiana, it loses 2.5 min each day. If the period of a pendulum of given length is proportional to 1/sqrt(g), and if g = 980.9 cm/s^2 in Paris, what is g in Cayenne?

...Um...I'm not entirely sure where to start. Do I just set the period, T = 1/sqrt(g)? If that's the case, how do I take into account the different of 2.5 minutes?

Answer 1

The difference of 2.5 minutes per day is assumed to be due to the difference in g at the two locations.

If in Cayenne it loses 2.5 minutes per day, it means the period there is longer.

To find how g affects the period, we are given the relationship T=k/sqrt(g) where k is a constant of proportionality determined by the equivalent length of the clock pendulum, but which we don't need to know.

Let
gp=g at Paris = 980.9 cm/s², and
gc=g at Cayenne in cm/s²,
then using the given relationship, we know that
T<paris>=k/sqrt(gp)
T<Cayenne)=k/sqrt(gc)

But we are also given the clock loses 2.5 minutes each day.
Each day has 24*60=1440 minutes, so losing 2.5 minutes a day means that the period at Cayenneis longer by a factor of
1440/(1440-2.5), which is our missing link.

We can now complete the equation above by saying

(k/√(gc))=(k/√(gp))*1440/1440-2.5
rearranging and cross-multiply, we get a simpler equation from which you can solve for gc:

√(gc)=(1437.5/1440)√(gp)