Thursday, November 15, 2012

Physics of a Lacrosse Stick: Shooting and Stick Length

In order to shoot 2 forces are involved:
Force from pulling stick backwards with bottom hand and force from pushing forward with the top hand- creating a lever arm (see figure below).

How does stick length affect the magnitude of force/work required for shooting the ball at 70 mph?

Measurements
Stick length regulation- 35.5 in-43.25 in (0.912 m-1.10m)
Stick: ~0.50 kg
Ball: ~0.14 kg;

Assumption:
The stick travels in a perfect circle in an overhand shot from the axis of rotation in center of the stick- (calculations ignores all momentum from body movement)
Assume stick is a uniformed rod and ball is a point mass
Assume shortest/longest stick has small mass
Forces are 0.1m away for the axis
Assume F are equivalent
Assume Δt= 0.3 seconds
Ball sits 0.05m from top of stick (r= 0.912-0.05= 0.862 m and 1.10-0.05 = 1.05 m)
No translational motion (only rotational)
Ball travels Δ θ= π/2 before released
Velocity of ball at the top of the shot (aka when released) = 70.0 mph (31.3 m/s)
Non-conservative forces are ignored (FAR, FG, etc.)

Calculations- shortest stick
v/r=Ѡ
Ѡ=(31.3 m/s)(0.912m-0.05m) = 27 rad/s
α = ΔѠ/ Δt
α = (27-0 rad/s)/(.3 s) = 90 rad/s2
Στ =Σ I Ѡ = ΣFr
Iball= mbr2 = (0.14 kg)(0.862 m)2= 0.104 kgm2
Istick= 1/12 msl2 = 1/12 (0.50 kg)(0.912 m)2 = 0.0347 kgm2
(0.104+0.0347 kgm2)(27 rad/s) = F (0.1 m) + F (0.1 m) = ΣF(0.2)
ΣF = 18.7 N
W= τ Δ θ
W= (3.74 mN) (π/2 rad) = 5.88 J

Calculations- longest stick
v/r=Ѡ
Ѡ=(31.3 m/s)(1.10m-0.05m) = 33 rad/s
α = ΔѠ/ Δt
α = (33-0 rad/s)/(.3 s) = 110 rad/s2
Στ =Σ I Ѡ = ΣFr
Iball= mbr2 = (0.14 kg)(1.05 m)2=  0.154 kgm2
Istick= 1/12 msl2 = 1/12 (0.50 kg)(1.10 m)2 = 0.0504 kgm2
(0.154+0.0504 kgm2)(33 rad/s) = F (0.1 m) + F (0.1 m) = ΣF(0.2)
ΣF = 33.7 N
W= τ Δ θ
W= (6.75 mN) (π/2 rad) = 10.6 J

THEREFORE, less force/input of work is required to shoot a ball at a certain speed with a shorter stick than a longer stick (it is no surprise then that attackers generally have shorter sticks than defenders).  But we have to remember in reality that is a lot more physic/biometrics to consider when determining the velocity and overall mechanics of a shot.

Here’s a video of the NCAA finals from last year.  You can check out some shots and see why these calculations above are insufficient: