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

L
m
r
4
Circular Motion

• The angle ? subtended by a distance s on the
  circumference of a circle of radius r

s
?
r
5
Radians

• Instead of measuring the angle ? in degrees (360
  to a circle), we can measure in pizza pi slices
  such that there are 2p 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 ? r with ? in
  radians.
• When a complete circle is traversed, ? 2p, and
  s 2p r, the circumference.

6
Angular Velocity

• When a wheel is rotating uniformly about its
  axis, the angle ? changes at a rate called ?,
  while the distance s changes at a rate called its
  velocity v.
• Then s r ? gives
• v r ?.

7
Angular Momentum and Moment of Inertia

• Lets recall the angular momentum
• L r m v r m (? r)
• L m r² ?
• In a rigid body, all parts rotate at the same
  angular velocity ?, so we can sum mr² over all
  parts of the body, to give
• I S mr², the moment of inertia of the body.
• The total angular momentum is then
• L I ?.

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 ?1 I2 ?2
• Example Rotating with weights out, pulling
  weights in shortens r, decreasing I and
  increasing ?.

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

• Bernoullis Equation (1738)
• Magnus Force (1852)
• Rayleigh Calculation (1877)

15
Bernoullis Principle

• Follow the flow of a certain constant volume of
  fluid ?V A?x, even though A and ?x change
• Pressure is PF/A
• Energy input is Forcedistance
• E F?x(PA)?xP?V
• kinetic energy is E½?v²?V
• So by energy conservation, P½?v² is a constant
• When v increases, P decreases, and vice-versa

?
16
Bernoullis Principal and Flight

• Lift on an airplane wing

V higher
P lower
P normal
v higher above wing, so pressure lower
17
Air around a rotating baseball, from balls 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
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