Sports and angular momentum

1. Sports and Angular
Momentum
Dennis Silverman
Bill Heidbrink
U. C. Irvine

2. 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

3. 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
m
r
L

4. Circular Motion
• The angle θ subtended by a distance s on
the circumference of a circle of radius r
r
θ
s

5. 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.

6. 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.

7. 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.

8. 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.

9. Examples of Moment of Inertia
• Hammer thrower
• Stick about different rotation axes
• Diver
• Baseball bat
• Pop quiz

10. 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.

11. 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

12. 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

13. 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

14. Curving of spinning balls
Bernoulli’s Equation (1738)
Magnus Force (1852)
Rayleigh Calculation (1877)

15. 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
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
Δ

16. Bernoulli’s Principal and Flight
• Lift on an airplane wing
v higher above wing, so pressure lower
P lower
P normal
V higher

17. Air around a rotating baseball, from
ball’s top point of view
Higher v, lower P on right
Lower v, higher P on left
So ball curves to right Pleft
Pright
Boundary layer

18. 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