Astronomers estimate that about 100 metric tons (1 ×10^5 kg) of meteorites strike the Earth ever

step 1 In this question we have given the moment of inertia of a body about a given axis and we have to

Astronomers estimate that about 100 metric tons (1 ×10^5 kg)...

Astronomers estimate that about 100 metric tons $\left(1 \times 10^{5} \mathrm{kg}\right)$ of meteorites strike the Earth every day. a. If the meteorites increased the mass of the Earth by $1 \%$ without adding any angular momentum, how long would the day become? b. The Earth's mass is $5.97 \times 10^{24}$ kg. How many years would it take to increase the Earth's mass by $1 \%$ at this rate?

Astronomers estimate that about 100 metric tons $\left(1 \times 10^{5} \mathrm{kg}\right)$ of meteorites strike the Earth every day.
a. If the meteorites increased the mass of the Earth by $1 \%$ without adding any angular momentum, how long would the day become?
b. The Earth's mass is $5.97 \times 10^{24}$ kg. How many years would it take to increase the Earth's mass by $1 \%$ at this rate?

Key Concepts

Mass Accretion Rate

The rate of mass accretion refers to how quickly a system gains mass over time. In this problem, understanding the steady daily addition of mass from meteorites allows us to estimate how long it would take for the accumulated mass to reach a given percentage of the Earth's total mass, thereby demonstrating the cumulative effect of small, regular contributions on a planetary scale.

Relationship Between Angular Velocity and Rotation Period

Angular velocity is inversely related to the period of rotation; that is, a decrease in angular velocity (while keeping angular momentum constant) leads to an increase in the rotation period. This relationship is key to understanding how the day lengthens when the moment of inertia increases.

Moment of Inertia

The moment of inertia is a measure of how a body's mass is distributed relative to its axis of rotation. When additional mass is added to the Earth at a greater average distance from the rotation axis, the moment of inertia increases, which, under conservation of angular momentum, results in a decrease in angular velocity and thus a longer rotation period or day.

Angular Momentum Conservation

This principle states that if no external torque acts on a system, its total angular momentum remains constant. In the context of the problem, when additional mass joins the Earth without contributing extra angular momentum, the Earth must adjust its rotational speed to maintain a constant angular momentum, leading to changes in the length of the day.

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