PX155 - E16 - using conservation of momentum

if no torques act about a point then the angular momentum is conserved about that point
ball striking a bat: no torque(or force) external to bat + ball $⟹ L$ is conserved
2 key points:
ball is in contact with the bat for a small time, $Δ T$ , during which, we ignore the movement of the bat
- ie: ball imports an impulse: $Δ p = \int F . d t$
after the impulse, no forces on the bat, so, $a_{c m} = 0$ for the bat. this means, the centre of rotation after the ball hits must be the centre of mass
conserve linear momentum:

m v_{c m} = Δ p

I_{c m} ω = (x - x_{c m}) Δ p

v_{H} = v_{c m} - ω x_{c m} = \frac{Δ p}{m} - \frac{(x - x_{c m}) Δ p c_{c m}}{I_{c m}}

x = x_{c m} + \frac{I_{c m}}{m x_{c m}}

uniform bat length, $L$ , $x_{c m} = \frac{L}{2}$ , $I_{c m} = \frac{1}{12} M L^{2} :$

x = \frac{2}{3} L