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Chapter 1: Q46E (page 332)

Asteroid Collision! Suppose that an asteroid traveling straight toward the center of the earth were to collide with our planet at the equator and bury itself just below the surface. What would have to be the mass of this asteroid, in terms of the earth鈥檚 mass M, for the day to become 25.0%longer than it presently is as a result of the collision? Assume that the asteroid is very small compared to the earth and that the earth and the earth are uniform throughout.

Short Answer

Expert verified

The mass of the asteroid that will make the day 25%longer is 0.133M.

Step by step solution

01

Given in the question.

Mass of earth is M.

Mass of asteroid is m.

Radius of earth is R

02

Conservation of angular momentum.

The law of conservation of angular momentum means that the angular momentum of a rotating system of objects always remains the same.

03

Moment of inertia of the earth.

For a day to be 25%longer, the angular velocity must be 25%lesser than the initial value. Hence,

蝇final=0.75蝇initial

The moment of inertia of the earth is calculated as follows:

Ii=25MR2

Here, Mis mass of earth and Ris the radius of earth

04

Moment of inertia of earth and asteroid.

The moment of inertia of earth and asteroid is given as:

If=25MR2+mR2=25M+mR2

Here, m is mass of asteroid.

05

Use conservation of angular momentum.

Use the conservation of angular momentum to obtain mass of asteroid:

Ii蝇i=If蝇f25MR2蝇i=25M+mR2蝇f25M蝇i=25M+m0.75蝇i25M=25M+m0.75

Solve further as follows:

25M-25脳34M=0.75mM10=0.75mM=7.5mm=M7.5m=0.133M

Hence, the mass of asteroid is0.133M.

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