Why is the following situation impossible? A meteoroid...

Why is the following situation impossible? A meteoroid strikes the Earth directly on the equator. At the time it lands, it is traveling exactly vertical and downward. Due to the impact, the time for the Earth to rotate once increases by 0.5 s, so the day is 0.5 s longer, undetectable to laypersons. After the impact, people on the Earth ignore the extra half-second each day and life goes on as normal. (Assume the density of the Earth is uniform.)

Key Concepts

Moment of Inertia of a Uniform Sphere

The moment of inertia quantifies a body's resistance to changes in its rotation rate. In a uniformly dense spherical body such as the theoretical Earth assumed in the problem, the moment of inertia is determined by its mass distribution relative to the axis of rotation. Without an appropriate redistribution of mass or application of tangential force, a purely vertical impact cannot alter the Earth’s moment of inertia in a manner that would lead to the described change in its rotational period.

Rotational Kinetics

This concept involves the relationship between angular velocity, torque, and moment of inertia. For a rotating body like the Earth, any alteration in the rotation period (or angular velocity) must be accounted for by changes in moment of inertia or the application of torque. Since the meteoroid in the scenario does not provide a tangential force component, its impact cannot yield the incremental slowdown needed to extend the day by a measurable half-second.

Conservation of Angular Momentum

This principle states that in the absence of an external torque, the total angular momentum of a system remains constant. In any collision or impact event involving a rotating body like the Earth, any change in the rotation period must come from a corresponding change in angular momentum. In the given problem, a meteoroid striking vertically should contribute negligible horizontal impulse, making it impossible to alter the Earth's rotational inertia without violating the conservation law.

Collision Dynamics and Impulse

Collisions are governed by the transfer of momentum and energy. An impact delivers impulse that can change the momentum of a system. In the scenario described, a meteoroid falling exactly vertically on the equator would impart no horizontal impulse or torque, rendering it incapable of affecting Earth’s rotation speed. This disconnect highlights why the stated change in the day length is physically inconsistent.

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