Momentum of an asteroid / Earth

[canvas01]

Please help me to understand this question: I'm paraphrasing it. A speed (relative to earth) and mass for an asteroid are given, and the asteroid strikes the Earth tangentially in the direction of the Earth's rotation at the equator. The question tells us to use angular momentum to find the percentage change in the angular speed of the earth.

My approach: Because the asteroid is not rotating, it has no angular momentum, just linear momentum represented as mv. The Earth however, has no linear momentum, just angular momentum which is equal to Iw where I = .4mr^2 and w = v/r where v = circumference of the earth/time of one complete rotation of the earth. I'm confused as to where to go from here? How are the angular momentum of the Earth and the linear momentum of the asteroid related?

 

[OlderDan]

Please help me to understand this question: I'm paraphrasing it. A speed (relative to earth) and mass for an asteroid are given, and the asteroid strikes the Earth tangentially in the direction of the Earth's rotation at the equator. The question tells us to use angular momentum to find the percentage change in the angular speed of the earth.

My approach: Because the asteroid is not rotating, it has no angular momentum, just linear momentum represented as mv. The Earth however, has no linear momentum, just angular momentum which is equal to Iw where I = .4mr^2 and w = v/r where v = circumference of the earth/time of one complete rotation of the earth. I'm confused as to where to go from here? How are the angular momentum of the Earth and the linear momentum of the asteroid related?

The asteroid does have angular momentum relative to the rotation axis of the earth. What is the definition of angular momentum? (It is not Iω; that is derived from the definition applied to a rotating rigid object)

 

[kyle8921]

To answer this question, we need to understand the concept of conservation of momentum. This principle states that the total momentum of a system remains constant unless acted upon by an external force.

In this scenario, the system consists of the asteroid and the Earth. Before the impact, the asteroid has a certain linear momentum due to its mass and speed, while the Earth has a certain angular momentum due to its mass and rotational speed. As the asteroid strikes the Earth tangentially, it transfers its linear momentum to the Earth, causing it to gain linear momentum in the same direction.

However, this also affects the Earth's angular momentum. As the asteroid's linear momentum is transferred to the Earth, it also causes a change in the Earth's rotational speed. This is because angular momentum is a product of both mass and rotational speed, and any change in either of these factors will result in a change in angular momentum.

To calculate the percentage change in the Earth's angular speed, we can use the equation:

% change in angular speed = (change in angular speed/original angular speed) x 100

In this case, the change in angular speed is caused by the transfer of linear momentum from the asteroid, and we can calculate it using the equation:

Change in angular speed = (linear momentum of asteroid/total moment of inertia of Earth) x original angular speed

Substituting the values given in the question, we can calculate the percentage change in the Earth's angular speed. This will give us a measure of how much the Earth's rotational speed has increased due to the impact of the asteroid.

In summary, the angular momentum of the Earth and the linear momentum of the asteroid are related through the principle of conservation of momentum. The transfer of linear momentum from the asteroid to the Earth causes a change in the Earth's angular momentum, which can be calculated using the equations mentioned above.