A uniform, thin rod hangs vertically at rest from a frictionless pin attached to its top end. The rod has a mass of $0\mathrm{.}580$ kg and a length of $1\mathrm{.}64$ m$.($Assume a coordinate system where the $+$y$-$direction is up and the $+$x$-$direction is to the right. The rod is free to swing about the pin in the x$-$y plane.$)$

$($a$)$

You take a baseball bat and strike the bottom end of the rod. At the instant the bat strikes, the force it applies to the rod is $(15\mathrm{.}7)$ N$.$ What is the acceleration $($in m$/$s$2)$ of the rod's center of mass at this instant? $($Express your answer in vector form.$)$

a $=$

m$/$s$2$

$($b$)$

What is the horizontal force $($in N$)$ that the pin exerts on the rod at this same instant? $($Express your answer in vector form.$)$

F $=$

N

$($c$)$

The rod then returns to hanging at rest. You again strike the rod with the bat, applying the same force, but now you strike it at its midpoint. What now is the acceleration of the center of mass $($in m$/$s$2)$ at the instant of impact? $($Express your answer in vector form.$)$

a $=$

m$/$s$2$

$($d$)$

What is the horizontal force $($in N$)$ that the pin exerts on the rod when the rod is struck at the midpoint? $($Express your answer in vector form.$)$

F $=$

N

$($e$)$

You wish to find a point on the rod where you can strike it with the bat so that the pin exerts no horizontal force on the rod. How far $($in m$)$ is this point from the top of the rod? This point is called the center of percussion.

m