The Earth has an angular speed of 7.272·10-5 rad/s in its rotation. Find the new angular speed if an asteroid (m = 3.45·1022 kg) hits the Earth while traveling at a speed of 4.41·103 m/s (assume the asteroid is a point mass compared to the radius of the Earth) in each of the following cases:

a) The asteroid hits the Earth dead center.

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b) The asteroid hits the Earth nearly tangentially in the direction of Earth's rotation.

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c) The asteroid hits the Earth nearly tangentially in the direction opposite of Earth's rotation.

To find the new angular speed of the Earth after the asteroid hits, we need to use the principle of conservation of angular momentum. Angular momentum is conserved when no external torque acts on a system.

The formula for angular momentum is: L = I * ω, where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.

Since the Earth is rotating about its axis, we can assume it has a moment of inertia equal to I = 2/5 * M * R^2, where M is the mass of the Earth and R is the radius of the Earth.

a) When the asteroid hits the Earth dead center, it hits along the axis of rotation. In this case, no torque is exerted on the Earth, so the angular momentum is conserved. Therefore, the initial angular momentum before the collision is equal to the final angular momentum after the collision.

Initially, the angular momentum is L_initial = I_initial * ω_initial = (2/5 * M * R^2) * ω_initial.

After the collision, the angular momentum is L_final = (2/5 * M * R^2) * ω_final.

Since angular momentum is conserved, we have L_initial = L_final, which can be written as: (2/5 * M * R^2) * ω_initial = (2/5 * M * R^2) * ω_final.

Canceling out the common terms, we get: ω_initial = ω_final.

Therefore, the new angular speed of the Earth after the asteroid hits dead center remains the same as the initial angular speed: ω_final = ω_initial = 7.272·10-5 rad/s.

b) When the asteroid hits almost tangentially in the direction of Earth's rotation, there is some external torque exerted on the Earth. The angular momentum is not conserved in this case.

To find the new angular speed, we need to consider the change in angular momentum.

The initial angular momentum is L_initial = I * ω_initial = (2/5 * M * R^2) * ω_initial.

After the collision, the final angular momentum is L_final = I * ω_final = (2/5 * M * R^2) * ω_final.

The change in angular momentum is ΔL = L_final - L_initial.

Since ΔL is equal to the torque (τ) exerted multiplied by the change in time (Δt), we can write: ΔL = τ * Δt.

The torque exerted on the Earth is given by τ = r * F, where r is the radius of the Earth and F is the force applied by the asteroid.

The force applied by the asteroid can be calculated using the formula F = m * v, where m is the mass of the asteroid and v is its velocity.

Substituting the values and rearranging the equation, we get: ΔL = (r * m * v) * Δt.

Since angular momentum is conserved, we have ΔL = 0.

Therefore, (r * m * v) * Δt = 0.

Since the time of collision (Δt) cannot be zero, we conclude that r * m * v = 0.

This means that for the asteroid to hit almost tangentially in the direction of Earth's rotation, the force exerted must be zero. This implies that the asteroid does not make any impact on the Earth's angular speed, and hence the new angular speed remains the same as the initial angular speed: ω_final = ω_initial = 7.272·10-5 rad/s.

c) When the asteroid hits almost tangentially in the direction opposite to Earth's rotation, the external torque exerted on the Earth is opposite to its initial angular momentum.

The initial angular momentum is L_initial = I * ω_initial = (2/5 * M * R^2) * ω_initial.

After the collision, the final angular momentum is L_final = I * ω_final = (2/5 * M * R^2) * ω_final.

The change in angular momentum is ΔL = L_final - L_initial.

Since ΔL is equal to the torque (τ) exerted multiplied by the change in time (Δt), we can write: ΔL = τ * Δt.

The torque exerted on the Earth is given by τ = r * F, where r is the radius of the Earth and F is the force applied by the asteroid.

The force applied by the asteroid can be calculated using the formula F = m * v, where m is the mass of the asteroid and v is its velocity.

Substituting the values and rearranging the equation, we get: ΔL = (r * m * v) * Δt.

Since angular momentum is conserved, we have ΔL = 0.

Therefore, (r * m * v) * Δt = 0.

Since the time of collision (Δt) cannot be zero, we conclude that r * m * v = 0.

This means that for the asteroid to hit almost tangentially in the direction opposite to Earth's rotation, the force exerted must be zero. This implies that the asteroid does not make any impact on the Earth's angular speed, and hence the new angular speed remains the same as the initial angular speed: ω_final = ω_initial = 7.272·10-5 rad/s.