Earth's day has increased by 0.002 s in the last century. If this increase in Earth's period is constant, how long will it take for Earth to come to rest?

Earth's Period Increase Calculation

To calculate how long it will take for Earth to come to rest if the increase in its period is constant, we need to first determine the rate of change of the Earth's period.

The increase in Earth's period over a century is $0.002$ seconds. Let's denote this increase as $\Delta t$.

Now, we need to find out how much time it will take for the Earth's period to increase by $1$ second, as this would represent the time it takes for the Earth's rotation to slow down to the point of coming to a stop.

If the increase in period is constant, then the rate of change of the period $\left( \frac{dP}{dt} \right)$ can be represented as:

$$ \frac{dP}{dt} = \frac{\Delta t}{\text{100 years}} $$

Given that $\Delta t = 0.002$ seconds and we want to find out the time it takes for the period to increase by $1$ second, we can set up the equation:

$$ \frac{1 \text{ second}}{\frac{dP}{dt}} = \frac{1 \text{ second}}{\frac{0.002 \text{ seconds}}{\text{100 years}}} $$

Solving this equation will give us the time it takes for Earth to come to rest:

$$ \text{Time} = \frac{1 \text{ second}}{\frac{0.002 \text{ seconds}}{\text{100 years}}} $$

$$ \text{Time} = \frac{100 \times 1}{0.002} \text{ years} $$

$$ \text{Time} = 50000 \text{ years} $$

So, it will take approximately 50,000 years for Earth to come to rest if the increase in its period continues to be constant.