Earth In Trouble?

Classical Mechanics Level 3

What will be the effect on the time period of the Earth's rotation about its axis if the ice at the north pole started melting and the water flowed towards the equator?

Increases Decreases Remain unchanged

2 solutions

Pranshu Gaba
Mar 18, 2016

Since there is no external torque on Earth, the angular momentum of Earth about its axis of rotation is conserved. The angular momentum of a system about an axis is given by $L = I\omega$ , where $I$ is the moment of inertia of the system about the given axis and $\omega$ is the angular velocity of the system.

Since the angular momentum is conserved, the initial momentum is equal to the final momentum.

$I_{i} \omega_{i} = I_{f} \omega_{f}$

As the ice melts, it moves from the pole to the equator and its distance from the axis of rotation increases. Recall that the greater the distance of mass from the axis, the greater is the moment of inertia. Therefore melting of ice increases the moment of inertia of Earth. Thus $I_{f} > I_{i}$ , which implies that $\omega_{f} < \omega_{i}$ .

Since the angular velocity of Earth decreases due to the melting of ice, the Earth now takes more time to complete one rotation. Thus $\boxed{\text{the time period of Earth's rotation increases}}$ $_\square$

I don't agree with the answer given to the question. Yes, using conservation of angular momentum is the main idea behind it but the density of ice and water also matters. Before ice melts, the concentration of mass is more near the equator and less near the poles because of the different densities of ice and water. But after ice melts, the density becomes same in both the regions(perhaps more in polar regions as density of water is highest at 4°C and temperature will be near it in those places) and now more mass will be concentrated near the poles as compared to before thus decreasing the moment of inertia and reducing the time period.

Archisman Bhattacharjee - 5 years, 2 months ago

Do you agree with me? If not then say what do you find wrong.

Archisman Bhattacharjee - 5 years, 2 months ago

The problems states that "the ice at the north pole started melting and the water flowed towards the equator". Since the ice is initially at the north pole, it does not contribute to the moment of inertia at all. After the ice melts and moves to the equator, it contributes to the moment of inertia, and therefore the net result is that Earth's moment of inertia increases.

Our system consists of only the ice, which melts and moves from the poles to the equator. The rest of the earth does not change, and we can exclude it from the system. Irrespective of the densities of ice and water, we see that mass moves away from the axis of rotation, therefore the moment of inertia increases.

Pranshu Gaba - 5 years, 2 months ago

Exactly! Nicely explained (+1)

Pulkit Gupta - 5 years, 2 months ago

Jase Jason
Mar 18, 2016

From what I know, waves of water push against the land which slows the earth's rotation. Therefore, if more water were to push against the earth's rotation, it would increase time, which means making it slower.

Earth In Trouble?

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