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The Vector Product and Torque • • • The torque vector lies in a direction perpendicular to the plane formed by the position vector and the force vector τ=rxF The torque is the vector (or cross) product of the position vector and the force vector More About the Vector Product • • The direction of C is perpendicular to the plane formed by A and B Direction is given by the right-hand rule • • • • fingers in direction of A curl into direction of B thumb gives direction of crossproduct C Be sure to use your right hand! The Vector Product Defined • • • • Given two vectors, A and B The vector (cross) product of A and B is defined as a third vector, C C is read as “A cross B” The magnitude of C is AB sin θ • θ is the angle between A and B Ch 11: Question 3 • Vector A is in the negative y direction, and vector B is in the negative x direction. What are the directions of • • AxB BxA 1 Ch 11: Question 2 • • Is the triple product, defined by A·(BxC) a scalar or a vector quantity? Explain why the operation (A·B)xC has no meaning. Torque and Angular Momentum • The torque is related to the angular momentum • Similar to the way force is related to linear momentum Angular Momentum • The instantaneous angular momentum L of a particle relative to the origin O is defined as the cross product of the particle’s instantaneous position vector r and its instantaneous linear momentum p More About Angular Momentum • • • The SI units of angular momentum are (kg. m2)/ s Both the magnitude and direction of L depend on the choice of origin The magnitude of L = mvr sin φ • • This is the rotational analog of Newton’s Second Law • • • φ is the angle between p and r The direction of L is perpendicular to the plane formed by r and p Στ and L must be measured about the same origin This is valid for any origin fixed in an inertial frame 2 Angular Momentum of a Rotating Rigid Object Angular Momentum of a Particle, Example • • The vector L = r x p is pointed out of the diagram The magnitude is L = mvr sin 90o = mvr • • Each particle of the object rotates in the xy plane about the z axis with an angular speed of ω • sin 90o is used since v is perpendicular to r A particle in uniform circular motion has a constant angular momentum about an axis through the center of its path The angular momentum of an individual particle is Li = mi ri2 ω L and ω are directed along the z axis • • 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 particles Angular Momentum of a Bowling Ball • • • This also gives the rotational form of Newton’s Second Law • The momentum of inertia of the ball is 2/5MR 2 The angular momentum of the ball is Lz = Iω The direction of the angular momentum is in the positive z direction 3 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 zero • • Conservation of Angular Momentum If the mass of an isolated system undergoes redistribution, the moment of inertia changes • • Net torque = 0 -> means that the system is isolated Ii ωi = If ωf Ltot = constant or • Li = Lf • Ch 11: Question 12 • Often when a high diver wants to turn a flip in midair, she draws her legs up against her chest. Why does this make her rotate faster? What should she do when she wants to come out of her flip? The conservation of angular momentum requires a compensating change in the angular velocity This holds for rotation about a fixed axis and for rotation about an axis through the center of mass of a moving system The net torque must be zero in any case Ch 11: Question 14 • Stars originate as large bodies of slowly rotating gas. Because of gravitation, these clumps of gas slowly decrease in size. What happens to the angular speed of a star as it shrinks? Explain. 4 Conservation of Angular Momentum: The Merry-Go-Round Conservation Law Summary For an isolated system (1) Conservation of Energy: • • • Ei = Ef (2) Conservation of Linear Momentum: • pi = pf (3) Conservation of Angular Momentum: • Li = Lf The moment of inertia of the system is the moment of inertia of the platform plus the moment of inertia of the person • • As the person moves toward the center of the rotating platform, the angular speed will increase • Ch 11: Question 13 • In some motorcycle races, the riders drive over small hills, and the motorcycle becomes airborne for a short time. If the motorcycle racer keeps the throttle open while leaving the hill and going into the air, the motocycle tends to nose upward. Why does this happen? Assume the person can be treated as a particle To keep L constant Ch 11: Question 15 • If global warming occurs over the next century, it is likely that some polar ice will melt and the water will be distributed closer to the Equator. How would this change the moment of inertia of the Earth? Would the length of the day increase or decrease? 5 Motion of a Top • • • • The only external forces acting on the top are the normal force n and the gravitational force M g The direction of the angular momentum L is along the axis of symmetry The right-hand rule indicates that τ = r × F = r × M g is in the xy plane Causes precession, slow rotation of top about z axis Angular Momentum as a Fundamental Quantity • • • The concept of angular momentum is also valid on a submicroscopic scale Angular momentum has been used in the development of modern theories of atomic, molecular and nuclear physics In these systems, the angular momentum has been found to be a fundamental quantity • • Fundamental here means that it is an intrinsic property of these objects It is a part of their nature Gyroscope in a Spacecraft • • • The angular momentum of the spacecraft about its center of mass is zero A gyroscope is set into rotation, giving it a nonzero angular momentum The spacecraft rotates in the direction opposite to that of the gyroscope so that the total momentum of the system remains zero. Can also use gyroscopes to keep a spacecraft pointed in a set direction (Hubble Space Telescope). Fundamental Angular Momentum • Angular momentum has discrete values • • • quantized These discrete values are multiples of a fundamental unit of angular momentum The fundamental unit of angular momentum is h-bar • Where h is called Planck’s constant 6 Classical Ideas in Subatomic Systems • Certain classical concepts and models are useful in describing some features of atomic and molecular systems • • Proper modifications must be made A wide variety of subatomic phenomena can be explained by assuming discrete values of the angular momentum associated with a particular type of motion Niels Bohr • • Niels Bohr was a Danish physicist He adopted the (then radical) idea of discrete angular momentum values in developing his theory of the hydrogen atom • Classical models were unsuccessful in describing many aspects of the atom Bohr’s Hydrogen Atom • The electron could occupy only those circular orbits for which the orbital angular momentum was equal to n h • • where n is an integer This means that orbital angular momentum is quantized 7