How Massive Must an Asteroid Be to Extend Earth's Day by 28%?

[SteamKing]

That's all fine and dandy, but the problem statement says that the size of the asteroid is neglegable. This is so you can treat it as a point mass. Since when do Mechanics 1 problems need to be realistic?

Turns out whoever wrote this problem either lied or didn't know what the answer was. Having something 10 times as massive as the moon hitting Earth is not 'negligible' in any sense of the word.

 

[BiGyElLoWhAt]

Ok, so then op needs to integrate r^2 dm over the density function of the asteroid (which we don't have) and sum that with the earth.
[edit] [earth]'s moment.

 

[SteamKing]

Ok, so then op needs to integrate r^2 dm over the density function of the asteroid (which we don't have) and sum that with the earth.

You can make a list of things we don't know about this asteroid, which is why this problem is poorly constructed.

 

[BiGyElLoWhAt]

This was just a problem that someone pulled out of their rear end. Think about it. An asteroid hits the Earth aimed at towards the center of the earth, and causes an increase in our day by 28% (!). That's huge for an asteroid that exerts 0 torque. It's all about recognizing you need to use angular momentum and applying it properly.

 

[haruspex]

28.0% longer than it presently is

BOAS, are you sure it's 28%, not perhaps 0.28%?

 

[mfb]

The asteroid is a point mass: "Assume that the asteroid is very small compared to the earth" - sure it is completely unrealistic, but we can ignore that issue.

Such a massive asteroid (Mars just has 10% of the mass of earth, our "asteroid" is larger!) would change the center of rotation of earth. This makes the new moment of inertia a bit smaller than calculated before.

Also, a day 28% longer than before does not correspond to a new angular velocity 28% smaller than before.

Putting all together, I get about 12.6% of the mass of Earth for the asteroid.

18.4% with the 28%-mistake, 11.2% with the wrong center of rotation and 15.56% with both at the same time.By the way, you do not need any data about Earth (apart from the constant density) - the result is true independent of its size, mass or length of a day.

 

[BOAS]

BOAS, are you sure it's 28%, not perhaps 0.28%?

It's definitely 28.0%.

 

[BOAS]

Also, a day 28% longer than before does not correspond to a new angular velocity 28% smaller than before.

Ah, you're right. it's 2pi radians in 1.28*86400 seconds