Question

Moment of inertia and planet's radius calculations.

A. (a) 1.22 × 10³⁸ kg·m²; (b) 4.0 × 10⁷ m
B. (a) 1.22 × 10³⁸ kg·m²; (b) 6.0 × 10⁷ m
C. (a) 9.74 × 10³⁷ kg·m²; (b) 4.0 × 10⁷ m
D. (a) 9.74 × 10³⁷ kg·m²; (b) 6.0 × 10⁷m

Answer 1

To calculate the change in angular velocity of Earth after a layer breaks off and the radius decreases, we can use the principle of conservation of angular momentum. We start by finding the initial and final moment of inertia using the formulas for a solid sphere.

Step-by-step explanation:

In this question, we are given information about Earth's mass and radius, as well as its angular velocity.

We are then asked to determine how the planet's angular velocity would change if a layer of Earth broke off and the radius decreased.

To solve this, we can use the principle of conservation of angular momentum.

Angular momentum is given by the equation L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

Since there is no friction, the initial angular momentum of the Earth is equal to the final angular momentum.

Therefore, we can set up the equation:

Initial angular momentum = Final angular momentum

Using the formula for the moment of inertia of a solid sphere and substituting the given values, we can determine the initial moment of inertia (I1) and the final moment of inertia (I2) as follows:

I1 = (2/5)MR^2

I2 = (2/5)M(R - r)^2

Finally, we can use the equation for conservation of angular momentum to calculate the change in angular velocity:

Δω = (I1/I2)ω1

where ω1 is the initial angular velocity.