How Does an Asteroid Impact Affect Earth's Angular Speed?

[FahimP]

Homework Statement

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 = 1.17 1022 kg) hits the Earth while traveling at a speed of 1.41 103 m/s. The asteroid hits the Earth dead center.

We are just learning physics so this is all basics... we are ignoring friciton right now. I suppose this is just a basic angular momentum conservaton problem but I am having difficulty understanding it.

The Attempt at a Solution

I used the angular momentum conservation and this is what i have. I got stuck at the last part and I'm pretty confident i am right so far but i don't know what's next . Please help / advice / tips
My problem is that there are too many unknowns and I'm not sure if anything cancels .. do I need a second Equation ?

Ra= Radius of asteroid
Re = radius of earth
Me = Mas of Earth
Ma = mass of asteroid
Wf = Final speed
We= Initial angular speed of Earth
Wa = initial angular speed of asteroid

Lao + Leo = Laf + Lef
MaRa^2 Wf + 2/5MeRe^2 Wf = MaRaVa + 2/5 MeRe^2We
Wf( MaR^2 + 2/5MeR^2) = MaVaRa + 2/5 MeRe^2We

 

[Mech_Engineer]

You have everything you need so think about it this way- at what incident angle does the asteroid hit the earth?

Assuming "dead center" means normal to the planet's surface (pointing to the imaginary center of the "spherical" planet), would its impact affect the Earth's rotation through momentum transfer, or simply add to the Earth's total mass (thus affecting its rotational inertia)?

 

[Oldblood]

You have everything you need so think about it this way- at what incident angle does the asteroid hit the earth?

Assuming "dead center" means normal to the planet's surface (pointing to the imaginary center of the "spherical" planet), would its impact affect the Earth's rotation through momentum transfer, or simply add to the Earth's total mass (thus affecting its rotational inertia)?

I have a question about this. When the asteroid hits the earth, won't the frictional force component during the impact generate a torque to the earth, thus slowing down its angular velocity, plus adding rotational inertia, slowing it further down? I am not sure about this, that's why I ask.

 

[Mech_Engineer]

I have a question about this. When the asteroid hits the earth, won't the frictional force component during the impact generate a torque to the earth,

The problem statement says the asteroid hits the Earth "dead center," which in my opinion precludes the possibility of the asteroid hitting the Earth at a tangent.

Of course this brings up another problem which I hadn't considered, WHERE on Earth does it hit? The effects will be different if it hits normal to the north pole versus at the equator, unless you make the assumption that the added mass can be approximated as being evenly distributed on the planet...

thus slowing down its angular velocity, plus adding rotational inertia, slowing it further down? I am not sure about this, that's why I ask.

This would all depend on which direction the asteroid hit. Assuming it hit at a tangent, it could either speed up rotation, slow it down, or tilt the rotational axis. Since according to the problem the asteroid is hitting normal to the Earth's surface, the effects should be seen in rotational inertia only, but it has occurred to me you need to make an assumption the added mass will be considered as evenly distributed around the "spherical" (also an assumption) planet.

In any case, it seems to me that the speed of the asteroid is irrelevant when you assume it hits the surface at a normal, and the mass is evenly distributed.