Today, as I was working on the Rotational Motion WebAssign, I remembered that if you drop a spinning basketball, it will bounce back up spinning in the opposite direction. I tried to wrap my head around it and hoped that application of some physics knowledge would reveal the odd phenomenon. So let's check out our basketball:
The ball has:
- an angular velocity ω
- a mass m (of 0.6kg!)
- a radius r (of 0.119m!)
- a moment of inertia i of 0.00569kg*m^2 (A basketball does have air in it but we'll assume it is hollow so i=(2/3)mr^2 )
Now you have a choice. Do you take the blue pill and live out the rest of your boring life? OR Do you take the red pill and dive down the rabbit hole?
If you have in fact entered the Matrix, imagine now that it's not the ball that spins. It's the ground! The ground is of course spinning about the same axis as the ball with an angular velocity of -ω
Now let's consider the earth. Thank you Kevin for the wealth of info! Relative to the ball, the ground has:
- an angular velocity Ω=-ω
- a mass of M 5.9742*10^24kg
- a radius of R 6.3781*10^6m
- a moment of inertia I of 9.7213*10^37kg*m^2 (from (2/5)MR^2 )
Now if we apply conservation of angular momentum:
Lo=Lf w/ L=Iω
To find the final angular velocities we have to find the (I think) the ω of the center of I as the rotational parallel of velocity of center of mass. So if vcm=Ptotal/Mtotal, ωci=Ltotal/Itotal.
ωci=((5.69*10^-3)(0)+(9.7213*10^37))(-ω)/(9.7213*10^37+5.69*10^-3) [because the earth is so massive and the ball is so relatively un-massive, I/(I+i) is just like I/I or 1]
From the ωci reference we can find the ωf by ωf=2ωci-ωo=2(-ω)-0=-2ω.
Also Ωf=2ωci-Ωo=2(-ω)-(-ω)=-ω. [Again this makes sense due to the earth's massive mass. A basketball is not going affect the earth in any great manner so Ω stays the same] Let's keep moving!
So after the collision, the ball, once at rest, is now rotating at -2ω. BUT WAIT! We're still in the Matrix. Back in the real world the ground is not moving. To get back to zero we have to add ω. Same goes for the basketball. And voila: -2ω+ω=-ω.
So the next time you spin a basketball with ω and it bounces back with -ω, be glad you took the red pill!