2 Angular MomentumIn analogy to the principle of conservation of linear momentum for an isolated system, the angular momentum of a system is conserved if no external torques act on the system.
3 The Vector ProductThere are instances where the product of two vectors is another vectorEarlier we saw where the product of two vectors was a scalarThis was called the dot productThe vector product of two vectors is also called the cross product
4 The Vector Product and Torque The torque vector lies in a direction perpendicular to the plane formed by the position vector and the force vectorThe torque is the vector (or cross) product of the position vector and the force vector
5 The Vector Product Defined Given two vectors, andThe vector (cross) product of and is defined as a third vector,C is read as “A cross B”The magnitude of vector C is AB sin qq is the angle between and
6 More About the Vector Product The quantity AB sin q is equal to the area of the parallelogram formed by andThe direction of is perpendicular to the plane formed by andThe best way to determine this direction is to use the right-hand rule
7 Properties of the Vector Product The vector product is not commutative. The order in which the vectors are multiplied is importantTo account for order, rememberIf is parallel to (q = 0o or 180o), thenTherefore
8 More Properties of the Vector Product If is perpendicular to , thenThe vector product obeys the distributive law
9 Final Properties of the Vector Product The derivative of the cross product with respect to some variable such as t iswhere it is important to preserve the multiplicative order of and
14 Torque Vector Example Given the force and location Find the torque produced
15 Angular MomentumConsider a particle of mass m located at the vector position and moving with linear momentumFind the net torqueWhy?Similar to the equation:
16 Angular Momentum, contThe instantaneous angular momentum of a particle relative to the origin O is defined as the cross product of the particle’s instantaneous position vector and its instantaneous linear momentum
17 Torque and Angular Momentum The torque is related to the angular momentumSimilar to the way force is related to linear momentumThe torque acting on a particle is equal to the time rate of change of the particle’s angular momentumThis is the rotational analog of Newton’s Second Lawmust be measured about the same originThis is valid for any origin fixed in an inertial frame
18 More About Angular Momentum The SI units of angular momentum are (kg.m2)/ sBoth the magnitude and direction of the angular momentum depend on the choice of originThe magnitude is L = mvr sin ff is the angle between andThe direction of is perpendicular to the plane formed by and
19 Angular Momentum of a Particle, Example A particle moves in the xy plane in a circular path of radius rFind the magnitude and direction of its angular momentum relative to an axis through O when its velocity is
20 Angular Momentum of a Particle, Example The linear momentum of the particle is changing in direction but not in magnitude.Is the angular momentum of the particle always changing?
21 Angular Momentum of a Particle, Example The vector is pointed out of the diagramThe magnitude isL = mvr sin 90o = mvrsin 90o is used since v is perpendicular to rA particle in uniform circular motion has a constant angular momentum about an axis through the center of its path
22 Angular Momentum of a System of Particles The total angular momentum of a system of particles is defined as the vector sum of the angular momenta of the individual particlesDifferentiating with respect to time
23 Angular Momentum of a System of Particles, cont Any torques associated with the internal forces acting in a system of particles are zeroTherefore,The net external torque acting on a system about some axis passing through an origin in an inertial frame equals the time rate of change of the total angular momentum of the system about that originThis is the mathematical representation of the angular momentum version of the nonisolated system model.
24 Angular Momentum of a System of Particles, final The resultant torque acting on a system about an axis through the center of mass equals the time rate of change of angular momentum of the system regardless of the motion of the center of massThis applies even if the center of mass is accelerating, provided are evaluated relative to the center of mass
25 System of Objects, Example A sphere of mass m1 and a block m2 are connected by a light cord that passes over a pulleyThe mass of the thin rim is M.M
26 System of Objects, Example The masses are connected by a light cord that passes over a pulley; find the linear accelerationConceptualizeThe sphere falls, the pulley rotates and the block slidesUse angular momentum approachM
27 System of Objects, Example cont CategorizeThe block, pulley and sphere are a nonisolated systemUse an axis that corresponds to the axle of the pulleyAnalyzeAt any instant of time, the sphere and the block have a common velocity vWrite expressions for the total angular momentum and the net external torqueML = m1vR + m2vR + MvR and
28 System of Objects, Example final Analyze, contSolve the expression for the linear accelerationSolve a!M
29 Angular Momentum of a Rotating Rigid Object Each particle of the object rotates in the xy plane about the z axis with an angular speed of wThe angular momentum of an individual particle is Li = mi ri2 wand are directed along the z axis
30 Angular Momentum of a Rotating Rigid Object, cont To find the angular momentum of the entire object, add the angular momenta of all the individual particlesThis also gives the rotationalform of Newton’s Second Law
31 Angular Momentum of a Rotating Rigid Object, final The rotational form of Newton’s Second Law is also valid for a rigid object rotating about a moving axis provided the moving axis:(1) passes through the center of mass(2) is a symmetry axisIf a symmetrical object rotates about a fixed axis passing through its center of mass, the vector form holds:where is the total angular momentum measured with respect to the axis of rotation
32 Angular Momentum of a Bowling Ball Estimate the magnitude of the angular momentum of a bowling ball spinning at 10 rev/s.
33 Angular Momentum of a Bowling Ball The momentum of inertia of the ball is 2/5MR 2The angular momentum of the ball is Lz = IwThe direction of the angular momentum is in the positive z direction
34 Conservation of Angular Momentum The total angular momentum of a system is constant in both magnitude and direction if the resultant external torque acting on the system is zeroNet torque = 0 -> means that the system is isolatedFor a system of particles,n
35 Conservation of Angular Momentum, cont If the mass of an isolated system undergoes redistribution, the moment of inertia changesThe conservation of angular momentum requires a compensating change in the angular velocityIi wi = If wf = constantThis holds for rotation about a fixed axis and for rotation about an axis through the center of mass of a moving systemThe net torque must be zero in any case
36 Conservation Law Summary For an isolated system -(1) Conservation of Energy:Ei = Ef(2) Conservation of Linear Momentum:(3) Conservation of Angular Momentum:
37 Conservation of Angular Momentum: The Merry-Go-Round A horizontal platform in the shape of a circular disk rotates freely in a horizontal plane about a frictionless vertical axle.The platform has a mass M=100kg and a radius R=2m.A student with mass 60kg walks slowly from the rim of the disk toward its center.If the angular speed of the system is 2 rad/s when the student is at the rim, what is the angular speed when she reaches a point r=0.5m from the center?
38 Conservation of Angular Momentum: The Merry-Go-Round The moment of inertia of the system is the moment of inertia of the platform plus the moment of inertia of the personAssume the person can be treated as a particleAs the person moves toward the center of the rotating platform, the angular speed will increaseTo keep the angular momentum constantWhy?
39 Conservation of Angular Momentum: The Merry-Go-Round Solve ωf
40 Conservation of Angular Momentum: The Merry-Go-Round What if you measured the kinetic energy of the system before and after the student walks inward?Are the initial kinetic energy and the final kinetic energy the
41 Conservation of Angular Momentum: The Merry-Go-Round The kinetic energy increases.The student must do work to move herself closer to the center of rotation.This extra kinetic energy comes from chemical potential energy in the student’s body.
42 Motion of a TopThe only external forces acting on the top are the normal force and the gravitational forceThe direction of the angular momentum is along the axis of symmetryThe right-hand rule indicates that the torque is in the xy plane
43 Motion of a Top The direction of is parallel to that of the top precesses about the z-axis
44 Motion of a Top, contThe net torque and the angular momentum are related:A non-zero torque produces a change in the angular momentumThe result of the change in angular momentum is a precession about the z axisThe direction of the angular momentum is changingThe precessional motion is the motion of the symmetry axis about the verticalThe precession is usually slow relative to the spinning motion of the top
45 Gyroscope A gyroscope can be used to illustrate precessional motion The gravitational force produces a torque about the pivot, and this torque is perpendicular to the axleThe normal force produces no torque
46 Gyroscope, contThe torque results in a change in angular momentum in a direction perpendicular to the axle.The axle sweeps out an angle df in a time interval dt.The direction, not the magnitude, of the angular momentum is changingThe gyroscope experiences precessional motionp: precessional frequency
47 Gyroscope, finalTo simplify, assume the angular momentum due to the motion of the center of mass about the pivot is zeroTherefore, the total angular momentum is due to its spinThis is a good approximation when is large
48 Precessional Frequency Analyzing the previous vector triangle, the rate at which the axle rotates about the vertical axis can be foundwp is the precessional frequencyThis is valid only when wp << w
49 Gyroscope in a Spacecraft The angular momentum of the spacecraft about its center of mass is zeroA gyroscope is set into rotation, giving it a nonzero angular momentumThe spacecraft rotates in the direction opposite to that of the gyroscopeSo the total momentum of the system remains zero
50 New Analysis Model 1 Nonisolated System (Angular Momentum) If a system interacts with its environment in the sense that there is an external torque on the system, the net external torque acting on the system is equal to the time rate of change of its angular momentum:
51 New Analysis Model 2 Isolated System (Angular Momentum) If a system experiences no external torque from the environment, the total angular momentum of the system is conserved:Applying this law of conservation of angular momentum to a system whose moment of inertia changes givesIiwi = Ifwf = constant