Sports and Angular Momentum
اسلاید 1: Sports and Angular MomentumDennis SilvermanBill HeidbrinkU. C. Irvine
اسلاید 2: OverviewAngular MotionAngular MomentumMoment of InertiaConservation of Angular MomentumSports body mechanics and angular momentumAngular Momentum and StabilityHow a baseball curves
اسلاید 3: Angular MomentumLinear momentum or quantity of motion is P = mv, and inertia given by mass m.m vRotation of a mass m about an axis, zero when on axis, so should involve distance from axis rAngular momentum L = r mvmrL
اسلاید 4: Circular MotionThe angle θ subtended by a distance s on the circumference of a circle of radius rrθs
اسلاید 5: RadiansInstead 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 circleSo 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.
اسلاید 6: Angular VelocityWhen 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.
اسلاید 7: Angular Momentum and Moment of InertiaLet’s recall the angular momentumL = r m v = r m (ω r) L = m r² ωIn 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 thenL = I ω.
اسلاید 8: Conservation of Angular MomentumIf 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.
اسلاید 9: Examples of Moment of InertiaHammer throwerStick about different rotation axesDiverBaseball batPop quiz
اسلاید 10: Applications of Conservation of Angular MomentumIf 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 ω2Example: Rotating with weights out, pulling weights in shortens r, decreasing I and increasing ω.
اسلاید 11: Examples of Changes in Moment of InertiaPulling arms in to do spins in ice skatingTucking while diving to do rollsBicycle wheel flip demoSpace station video
اسلاید 12: Rotating different parts of bodyBallet pirouetteBalancing beamIce skater balancingFalling cat or rabbit landing uprightRodeo bull riderSki turnsSki jumping video
اسلاید 13: Angular Momentum for StabilityBicycle or motorcycle ridingFootball pass or lateral spinningSpinning topFrisbeeSpinning gyroscopes for orbital orientationHelicopterRifling of rifle barrelEarth rotation for daily constancy and seasons
اسلاید 14: Curving of spinning ballsBernoulli’s Equation (1738)Magnus Force (1852)Rayleigh Calculation (1877)
اسلاید 15: Bernoulli’s PrincipleFollow the flow of a certain constant volume of fluid ΔV =A*Δx, even though A and Δx changePressure is P=F/AEnergy input is Force*distance E = F*Δx=(PA)*Δx=P*ΔV kinetic energy is E=½ρv²ΔVSo by energy conservation, P+½ρv² is a constantWhen v increases, P decreases, and vice-versaΔ
اسلاید 16: Bernoulli’s Principal and FlightLift on an airplane wingv higher above wing, so pressure lowerP lowerP normalV higher
اسلاید 17: Air around a rotating baseball, from ball’s top point of viewHigher v, lower P on rightLower v, higher P on leftSo ball curves to rightPleftPrightBoundary layer
اسلاید 18: Examples of curving ballsBaseball curve pitchBaseball outfield throw with backspin for longer distanceTennis topspin to keep ball downSoccer (Beckham) curve around to goalGolf ball dimpling and backspin for rangeDeflection d = ½ a t² most at end of range
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