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?
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)
To solve this problem, we need to understand the relationship between the period of a pendulum and the acceleration due to gravity. The period of a pendulum, T, is given by the formula T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity.
In this problem, we are given that the period of a pendulum of a given length is proportional to 1/√(g). We are also given that the period of the pendulum clock changes by 2.5 minutes each day when it is moved from Paris to Cayenne.
First, we need to determine the initial value of g in Paris. We are told that g = 980.9 cm/s^2 in Paris. We can use this value to find the initial period of the pendulum clock.
T1 = 2π√(L/g1)
Next, we need to determine the final value of g in Cayenne. We can use the fact that the period of the pendulum clock changes by 2.5 minutes each day to find the change in g.
ΔT = T2 - T1 = 2.5 minutes
To find the change in g, we rearrange the equation for the period of the pendulum:
T = 2π√(L/g)
Square both sides of the equation:
T^2 = (2π)^2(L/g)
Rearrange to solve for g:
g = (2π)^2(L/T^2)
Since we know that T1 = 2π√(L/g1) and T1 + ΔT = 2π√(L/g2), we can substitute these values into the equation to find g2:
g2 = (2π)^2(L/((2π√(L/g1) + ΔT)^2))
Now, we can substitute the given values into the equation to find g2.
g1 = 980.9 cm/s^2
ΔT = 2.5 minutes = 2.5 * 60 seconds
L (length of the pendulum) is not given, so we can leave it as a variable for now.
By substituting these values into the equation, we can calculate g2 as follows:
g2 = (2π)^2(L/((2π√(L/980.9) + 2.5*60)^2))
Calculating this expression will give us the value of g in Cayenne, which is the final answer to the question.