The Earth’s rotation is indeed being slowed down by the presence of the Moon - every year, the Moon gains a little energy from the Earth, and drifts a little farther away from us. This drift is imperceptible to the human eye, but measurable, with the aid of undertakings like the Lunar Laser Ranging Experiment, which regularly bounces a laser off of a retroreflector that Apollo astronauts placed there.
Both the drift of the Moon and the slowing of the rotation of the Earth are very very small effects- the slowing of the Earth’s rotation over the last 100 years is estimated to be about 1.4 milliseconds. That’s a slowing of 0.0014 seconds total, over 100 years. Another method of estimating the slowing of the Earth uses historical records of solar eclipses to figure out exactly how fast the Earth must have been rotating in the past, and comes up with an average slowing of 2.5 milliseconds each century. To extrapolate out into the future, I’m going to use the average of these two numbers, and guess that we’re dealing with a slowing of approximately 0.002 seconds every century.
As a point of reference, this rate of slowing means that it will take 25,000 years to add a half a second to the Earth’s day. A whole second will take 50,000 years.
To add an entire hour? Every hour contains 3,600 seconds - (60 minutes to an hour, and 60 seconds to a minute). And so, to wait long enough to gain 3,600 seconds, we’ll need to wait 50,000 years 3,600 times over - 180 million years.
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