Cross products- how to determine the angle?

[wu7]

Consider the following problem:
You strike a pool ball (radius R) at some height h above the table, so that it immediately rolls without slipping.
At what height h should you strike it?

I worked out the problem and got 1.4R, which is indeed the answer.

However, at one point I made the assumption that I₀ω = Δp*(h-R)*sin90.
As in, the momentum is perpendicular to the radius, so the cross product is just times 1.

I don't understand why sin90 is acceptable. By that logic, you could hit the pool ball anywhere horizontally, and it would be perpendicular to some R. what R? In this case it seems to be the R which is vertical.

I thought you had to take the cross product at the point of contact- and use the radius which goes from the center to that point of contact. THEN cross that with the horizontal force.

I know this is the way you're supposed to do it with a rotating stick- why not with a sphere?

 

[jbsteele]

The answer I got is correct, but I'm wondering why my assumption is acceptable.The reason why your assumption is acceptable is that the momentum of the ball is perpendicular to its radius. Therefore, no matter where you strike it, the momentum is always perpendicular to the radius of the ball. Thus, the cross product of the momentum and the radius of the ball will always be the same, so you can just simplify it to sin90. The point of contact isn't really relevant in this case.