A baseball bat has a \"sweet spot\" where a ball can be hit with almost effortle

Question

A baseball bat has a "sweet spot" where a ball can be hit with almost effortless transmission of energy. A careful analysis of baseball dynamics shows that this special spot is located at the point where an applied force would resu

lt in pure rotation of the bat about the handle grip.

Determine the location of the sweet spot of the bat shown in the figure . The linear mass density of the bat is given roughly by (0.61+3.3x2)kg/m, where x is in meters measured from the end of the handle. The entire bat is 0.85m long. The desired rotation point should be 5.0 cm from the end where the bat is held. [Hint: Where is the cm of the bat?]

Explanation / Answer

It's called the "center of percussion".

A force at that location is going to give a torque about the center of mass, which will cause the bat to rotate about the center of mass. In the picture, if you rotate the right side upward, the handle will rotate downward.

But at the same time, a force causes the entire bat to move forward by F = ma. This causes the handle to move upward also.

The COP is the point that causes these effects to cancel at the handle.

Let r = distance of COP from center of mass. So the torque equals F*r, and this causes acceleration F*r = I * dw/dt where I = moment of inertia, w = angular velocity. The grip, the point on the left where you are holding the bat, is at distance r1 = (d - r) from the COM. So angular velocity w causes linear velocity w/r1 at that point (downward), and

F*r = (I/r1)*dv/dt

dv/dt = F*r*r1/I

The forward motion is F = ma = m*dv/dt, or dv/dt = F/m. You want these two dv/dt's to be equal, so

F*r*r1/I = F/m

The distance r1 is fixed since the COM is fixed. r is the free variable. Solving for r

r = I * r1/m

So here's one approach:
1. Find the total mass m by integrating (0.61 + 3.3x^2) over the length of the bat.

2. Find the center of mass by integrating x*dm = (0.61 + 3.3x^2) * x and dividing by total mass m.

3. Find the moment of inertia by integrating (0.61 + 3.3x^2) * (distance from COM)^2.

4. Figure out what r1 is.

5. Plug those into the formula above to get r.

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