Angular Impulse/Momentum
for Rigid Bodies in a Plane(example continued)
Use
Principle of Work and Energy to calculate the angular speed just prior to
impact.The dotted representation of
the bat designates position 1 where the bat starts from rest, ω1 =
0.The solid representation of the
bat designates position 2 where the bat is about to strike the baseball.In position 2,θ = π/6 radians (30o).Thus the bat swings thru a total arc of
2π/3 radians to the point of impact.
Apply:
W1-2=T2-T1
Principle of
Work/Energy
The
work done on the bat, W1-2, =Mo∆θ =Mo(θ+π/2)=2π/3 Mo
The
change in kinetic energy, T1-2, = ½ IzzO
ω22 – ω12 =½ (1/3 ML2) ω22=(1/6) (ML2) ω22
Set
work equal to change in KE:2π/3 Mo= (1/6) (ML2) ω22orω22=4π Mo / ML2
Soω2= √ [4π Mo /ML2 ]for the data:Mo = 50 ft lb,M = (32/16g)slugs,L = 3 ft
The result is the angular speed just
prior to impactω2=33.5 rad/sec
To
analyze the situation at impact construct a free body diagram showing the
impulsive forces acting on the bat and on the ball.
