Sports and Angular
Momentum
Dennis Silverman
Bill Heidbrink
U. C. Irvine
Overview
Angular Motion
Angular Momentum
Moment of Inertia
Conservation of Angular Momentum
Sports body mechanics and angular
momentum
Angular Momentum and Stability
How a baseball curves
Angular Momentum
 Linear momentum or quantity of motion is
P = mv, and inertia given by mass m.
 mv
 Rotation of a mass m about an axis, zero
when on axis, so should involve distance
from axis r
 Angular momentum L = r mv
L
r
m
Circular Motion
• The angle θ subtended by a distance s on
the circumference of a circle of radius r
s
θ
r
Radians
• Instead of measuring the angle θ in degrees
(360 to a circle), we can measure in pizza pi
slices such that there are 2π = 6.28 to a full
circle
• So each radian slice is about a sixth of a circle
or 57.3 degrees.
• Then we can write directly: s = θ r with θ in
radians.
• When a complete circle is traversed, θ = 2π, and
s = 2π r, the circumference.
Angular Velocity
• When a wheel is rotating uniformly about
its axis, the angle θ changes at a rate
called ω, while the distance s changes at a
rate called its velocity v.
• Then s = r θ gives
•
v = r ω.
Angular Momentum and
Moment of Inertia
•
•
•
•
Let’s recall the angular momentum
L = r m v = r m (ω r)
L = m r² ω
In a “rigid body”, all parts rotate at the same
angular velocity ω, so we can sum mr² over all
parts of the body, to give
• I = Σ mr², the moment of inertia of the body.
• The total angular momentum is then
• L = I ω.
Conservation of Angular
Momentum
• If there are no outside forces acting on a
symmetrical rotating body, angular momentum is
conserved, essentially by symmetry.
• The effect of a uniform gravitational field cancels
out over the whole body, and angular
momentum is still conserved.
• L also involves a direction, where the axis is the
thumb if the motion is followed by the fingers of
the right hand.
Examples of Moment of Inertia
•
•
•
•
•
Hammer thrower
Stick about different rotation axes
Diver
Baseball bat
Pop quiz
Applications of Conservation of
Angular Momentum
• If the moment of inertial I1 changes to I2 ,
say by shortening r, then the angular
velocity must also change to conserve
angular momentum.
• L = I1 ω1 = I2 ω2
• Example: Rotating with weights out,
pulling weights in shortens r, decreasing I
and increasing ω.
Examples of Changes in
Moment of Inertia
•
•
•
•
Pulling arms in to do spins in ice skating
Tucking while diving to do rolls
Bicycle wheel flip demo
Space station video
Rotating different parts of body
•
•
•
•
•
•
•
Ballet pirouette
Balancing beam
Ice skater balancing
Falling cat or rabbit landing upright
Rodeo bull rider
Ski turns
Ski jumping video
Angular Momentum for Stability
•
•
•
•
•
•
•
•
Bicycle or motorcycle riding
Football pass or lateral spinning
Spinning top
Frisbee
Spinning gyroscopes for orbital orientation
Helicopter
Rifling of rifle barrel
Earth rotation for daily constancy and seasons
Curving of spinning balls
Bernoulli’s Equation (1738)
Magnus Force (1852)
Rayleigh Calculation (1877)
Bernoulli’s Principle
• Follow the flow of a certain constant volume of
fluid ΔV =A*Δx, even though A and Δx change
• Pressure is P=F/A
• Energy input is Force*distance
Δ
E = F*Δx=(PA)*Δx=P*ΔV
• kinetic energy is E=½ρv²ΔV
• So by energy conservation, P+½ρv² is a
constant
• When v increases, P decreases, and vice-versa
Bernoulli’s Principal and Flight
• Lift on an airplane wing
V higher
P lower
P normal
v higher above wing, so pressure lower
Air around a rotating baseball, from
ball’s top point of view
Higher v, lower P on right
Pright
Boundary layer
Lower v, higher P on left
So ball curves to right
Pleft
Examples of curving balls
Baseball curve pitch
Baseball outfield throw with backspin for
longer distance
Tennis topspin to keep ball down
Soccer (Beckham) curve around to goal
Golf ball dimpling and backspin for range
Deflection d = ½ a t² most at end of range