Question

Because Earth’s rotation is gradually slowing, the length of each day increases: The day at the end of 1.0 century is 1.0 ms longer than the day at the start of the century. In 20 centuries, what is the total of the daily increases in time?

Short answer

The total of the daily increases in time is 2 hours.

Step-by-step solution

Calculating the average time of daily increase.

It is the angular velocity of the Earth that determines its rotation. Similarly, the length of each day is the time it takes to complete one rotation. Velocity is the rate of change of displacement with respect to time.

The day at the end of the 1.0 century is 1.0 ms longer than the start of the century.

The last day of the 20^th century is longer than the first day by,

$20\mathrm{century}\left(\frac{0.001\mathrm{s}}{1\mathrm{century}}\right)=0.02\mathrm{s}$

Average time of daily increase in 20 centuries from the first day is

$\frac{0+0.02}{2}=0.01s$

Calculating the total time increase in 20 centuries.

Total time increase in 20 centuries is

$$\begin{gathered} \mathrm{T}=\left(\frac{0.01\mathrm{s}}{1\mathrm{day}}\right)\left(\frac{365.25\mathrm{days}}{1\mathrm{y}}\right)\left(\frac{100\mathrm{y}}{1\mathrm{century}}\right)20\mathrm{century} \\ =7305\mathrm{s} \end{gathered}$$

Convert the time from seconds to hours.

$$\begin{gathered} \mathrm{T}=\frac{7305}{3600} \\ =2.03\mathrm{hrs} \\ \approx 2\mathrm{hrs} \end{gathered}$$

Thus, for 20 centuries daily increase in time is .2 hrs