What must x be so that the handle end of the bat remains at rest as the bat begins to move? (Hint: Consider the motion of the center of mass and the rotation about the center of mass. Find x so that these two motions combine to give v=0 for the end of the bat just after the collision. Also, note that integration of equation ∑τ⃗ =dL⃗ dt gives ΔL=∫t1t2(∑τ)dt. )

Complete Question

The complete question is shown on the first uploaded image

Explanation:

From the question we are told that

     The mass of the bat is

      The bat length is  

      The distance of the bat's center of mass to the handle end is  

      The moment of inertia of the bat is    

The objective of the solution is to find  x   which is the distance from the handle of the bat to the point where the baseball hit the bat

Generally the velocity change at the end of the bat is mathematically represented as

         Where   is the velocity change at the center of the bat  which is mathematically represented as

We are told that the impulse is  J so

And   is the change in angular velocity which is mathematically represented as

Now we have that

Before a swing when the bat is at rest the velocity change a the end of the bat handle is zero  and the impulse will be  1

   So  

=>          

substituting values