Sports and Angular Momentum

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