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 m r
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 θin 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 = I1ω1 = I2ω2 • 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 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