EPT
But anyway, I've been on fall break until now and classes resume tomorrow, and I have a physics problem set due in 13 hours that I'm having trouble on. Below is the physics problem and what I have so far. I have most of it done, but I can't finish it. It's an introductory physics course (PHY 53), so it shouldn't be too difficult.
82. 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 result in pure rotation of the bat about the handle grip. Determine the location of the sweet spot of the bat shown below. The linear mass density of the bat is given by (0.61 + 3.3 x^2) kg/m, where x is in meters measured from the end of the handle. The entire bat is 0.84m long. The desired rotation point should be 5.0 cm from the end where the bat is held.
![[image loading]](http://img523.imageshack.us/img523/5319/physicswz2.jpg)
Ok, so far, I know there are equal and opposite impulses acting on where the ball hits and on the batter's hand. So MΔv = -J - J'.
J' refers to impulse at the pivot point, J refers to impulse at where the ball hits.
Also, IΔω = angular impulse = -Jd. Also, v = d(CM)ω.
So, Md(CM)Δω= -J - J'.
Thus,
J' = -(1 - (M*d(CM)*d)/I)*J.
So we want to make M*d(CM)*d equal to I, the moment of inertia, so the term is always 0 and thus there is no impulse J' applied to the pivot point. So the sweet spot is where the moment of inertia equals the mass times d(CM) times d.
My problem is, they're looking for a numerical solution. I know I have to find where the center of mass is and the moment of inertia. Problem is, I don't know how. I'm thinking moment of inertia is found with the parallel axis theorem, and I think it might be (1/12)Ml^2 + M(r)^2, but I'm not sure what I'm supposed to put for r. Also I'm a little confused on how to find M from the linear mass density of the bat (because I'm stupid and forgot how to do this from high school).
Finally, I'm not sure how to calculate the center of mass. I think you just integrate along the bat, but again, I'm stupid and forgot how to do this from high school.
So yeah, rare case in which I have the theoretical solution but not the numerical solution.
Any help is appreciated...though after thirteen hours, I'll probably delete this blog post since by then my problem set'll've been due and I probably'll've missed some points on this problem.
cm*alpha ----------------------------------------------- (1)
