Chapter 12, Problem 36A

If the polar ice caps of Earth were to melt, the oceans everywhere would be deeper by about 30 m. What effect would this have on Earth’s rotation?

To determine

To explain:The effect of polar ice caps on Earth’s rotation.

Answer to Problem 36A

Earth would slow in its daily rotation and the day would become longer.

Explanation of Solution

Given:

The oceans will be at a depht of 30 m if polar ice caps of Earth will melt.

Formula used:

The expression for the angular momentum is,

  $L=mvr$

Here, $m$ is the mass, $v$ is the tangential speed, and $r$ is the radius of the curved path.

The expression for the velocity is,

  $v=r\omega$

Here, $r$ is the radius of the curved path and $\omega$ is the angular velocity.

The expression for the rotational speed is,

  $\omega =\frac{2\pi }{T}$

Here, $T$ is the time period.

Calculation:

The angular momentum is,

  $\begin{array}{l}L=mvr\\ L=m\left(r\omega \right)r\\ L=m{r}^2\omega \end{array}$

Consider that the angular momentum is conserved, the rotational speed is inversely proportional to the mass.

  $\omega =\frac{1}{m{r}^2}$

The Earth’s rotation depends on its mass at its equator.As the polar ice caps of the Earth melts, its mass spreads evenly along the equator of the Earth.The Earth’s rotational speed decreases due to the increase in Earth’s mass along the equator of the Earth.

The rotational speed is,

  $\omega =\frac{2\pi }{T}$

The rotational speed and the time period are inversely proportional to each other.Decrease in Earth’s rotational speed increases its time period of the rotation.Hence, Earth would slow in its daily rotation and the day would become longer.

Conclusion:

Thus, Earth would slow in its daily rotation and the day would become longer.