Calculating Earth Minutes on the Moon with a Grandfather Clock

Question:

In the year 2125, when we have a colony on the moon, an engineer brings an old grandfather clock with her. She knows the clock's pendulum has a length of 1.0 m and the moon's gravity is 1.62 m/s^2. If she winds the clock when the time shows 12:00, how many Earth minutes have elapsed when the clock face reads 12:33?

Answer:

0.047 Earth minutes have elapsed when the clock face reads 12:33.

Calculating Earth Minutes on the Moon with a Grandfather Clock

In order to determine how many Earth minutes have elapsed when the grandfather clock on the moon reads 12:33, we need to first calculate the time it takes for the pendulum to complete one full swing, or period, on the moon.

The formula for the period of a pendulum is T = 2π√(L/g), where T is the period in seconds, L is the length of the pendulum in meters, and g is the acceleration due to gravity in m/s^2. Plugging in the values given for the length of the pendulum and the moon's gravity, we get:

T = 2π√(1.0/1.62) = 4.59 seconds

Therefore, the clock on the moon will tick once every 4.59 seconds.

Next, we need to determine how many ticks of the clock have occurred between 12:00 and 12:33 on the moon. There are 33 minutes between 12:00 and 12:33, or 33 x 60 = 1980 seconds. Dividing 1980 by 4.59 gives us:

1980/4.59 ≈ 431.38

So, approximately 431 ticks of the clock have occurred between 12:00 and 12:33 on the moon.

Finally, we need to convert the number of seconds on the moon to Earth minutes. Since the moon's day is approximately 29.5 Earth days long, one Earth minute is equivalent to 29.5 x 24 x 60 = 42,480 moon seconds. Dividing 431 x 4.59 by 42,480 gives us:

431 x 4.59/42,480 ≈ 0.047 Earth minutes

Therefore, the clock on the moon will show 12:33 on Earth approximately 0.047 minutes after it was wound at 12:00.