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 Period of the Pendulum: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.
Calculating Number of Clock Ticks: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.
Converting to Earth Minutes: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.