Centre of percussion of a baseball bat

[Clara Chung]

Homework Statement

Homework Equations

The Attempt at a Solution

I don't know how to do the last part. How can there be no change in velocity of there is a change in angular velocity of the bat? Is the grip point acting as another axis?

 

[haruspex]

How can there be no change in velocity of there is a change in angular velocity of the bat?

If the linear velocity of the mass centre changes and the angular velocity changes then there will be a unique point at which these two cancel, leaving no change in linear velocity.

 

[Clara Chung]

If the linear velocity of the mass centre changes and the angular velocity changes then there will be a unique point at which these two cancel, leaving no change in linear velocity.

I tried to express change in angular velocity, which is
MΔv=J and Mk₀² Δw = (d+1.3)J
Δv+Δw=0
so I get,
J/M + (d_grip+0.8)(d+1.3)J/ Mk₀² = 0
Why can't I get the answer ?

 

[haruspex]

(d+1.3)J

Where does the 1.3 come from? What axis are you using here?

Δv+Δw=0

You cannot add a velocity to an angular velocity.
You are trying to express the fact that a certain point does not change linear speed. How does its speed depend on the speed of the mass centre and its distance from the mass centre?
Be careful with signs.

 

[Clara Chung]

Where does the 1.3 come from? What axis are you using here?

You cannot add a velocity to an angular velocity.
You are trying to express the fact that a certain point does not change linear speed. How does its speed depend on the speed of the mass centre and its distance from the mass centre?
Be careful with signs.

I am using the rotational pivot. Am I right?

 

[haruspex]

I am using the rotational pivot. Am I right?

Does that fit with the way k₀ is defined?

 

[Clara Chung]

Does that fit with the way k₀ is defined?

Thank you so much. I solved it by using the centre of mass frame^^

 

[haruspex]

Thank you so much. I solved it by using the centre of mass frame^^

Very good.
For problems involving angular momentum, you should always use an axis that is either a point fixed in an inertial frame or the mass centre. Anything else can lead to a wrong result.