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 m r L

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

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 I 1 changes to I 2, say by shortening r, then the angular velocity must also change to conserve angular momentum. L = I 1 ω 1 = I 2 ω 2 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 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

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

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 P left P right Boundary layer

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