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Chapter 8 More on Angular Momentum and Torque Main Points of Chapter 8 • Any moving object can have angular momentum around a fixed point • Angular momentum and torque can be expressed as vector products • Angular momentum is conserved • Precession 8-1 Generalization of Angular Momentum We can give the definition of angular momentum to a particle (with respect to the point O) S O Discussion • Angular momentum of a particle depends on both its momentum and position vector (on the point O) • We find that the angular momentum is constant for an object moving at constant velocity. 8-2 Generalization of Torque We can write an analogous expression for the torque with respect to the point O O h Fr sin Fh P The torque depends on both the force and the reference point O The relation between the torque and the angular momentum of a particle with respect to the same point O This is the rotational version of Newton’s second law 8-3 The Dynamics of Rotation For system of particles dL dt dL dt • Torque and angular momentum can be measured from (the same) any reference point • The angular momentum of a system of particles can be changed only by the external torques. • The angular momentum of a particle in the system can be changed by the internal torques. dLcm • In center-of-mass reference frame, cm dt holds even center-of-mass is accelerating. For a rigid body (rotating about a fixed axis) z dLz z dt I z Rigid O body fixed axis ri vi P Angular impulse 1. for system of particles angular impulse The change in angular momentum of system of particles during a time interval equals the angular impulse that acts on the particles during that interval 2. for a rigid body (rotating about a fixed axis) The angular impulse acts on the a rigid body rotating about a fixed axis equals to the change of angular momentum about the same axis. Example A slender uniform rod of length l and mass m is initially lying on a frictionless horizontal surface. A horizontal impulse FDt exerts on it at a distance d from C , describe the subsequent motion of the rod. Solution Consider CM motion d c FDt Consider rotation motion about CM CAI 8-4 Conservation of Angular Momentum If the net external torque acting on a system is zero, the angular momentum of the system remains constant, no matter what changes take place within the system. 1. For a particle 0 L Constant Vector a central force For central forces, angular momentum around origin is conserved and motion is in a plane L This leads to Kepler’s second law: The radius vector of a planetary orbit sweeps out equal areas in equal times. dr m r Example A space vehicle is launched to a planet with mass M, radius R, when the space vehicle reaches the planet at a distance of 4R, it launches an apparatus with mass m at velocity v0. If this apparatus can even sweep the surface of the planet, find the speed of landing and angularθ m v0 Solution: conservation of mechanical energy and angular momentum v R r0 OM 1 GMm 1 GMm 2 2 mv 0 mv 2 r0 2 R mv 0 r0 sin mvR 1/ 2 1 3GM sin 1 2 4 2 Rv 0 1/ 2 3GM v v 0 1 2 2 Rv 0 Act A puck with mass m slides in a circular path on a horizontal frictionless table. It is held by a string thread through a frictionless hole at the center of the table. If you pull on the string such that the radius decreases from r0 to r1, How much work have you done? Solution: r0 r1 Conservation of Angular Momentum mv 0 r0 mvr r0 v v0 r1 v0 v0 F 1 1 2 mv 1 mv 02 2 2 2 2 r r 1 1 1 W mv 02 0 mv 02 mv 02 0 1 2 2 r1 r1 2 Work-Energy theorem W Example a particle of mass m is shooting horizontally in a hemispherical bowl which is frictionless. The particle can just reach the rim of bowl. Find v 0 Solution: conservation of mechanical energy 1 1 2 mv 0 mv 2 mgrCos 0 2 2 z o N v0 mg conservation of angular momentum in z direction (Z=0) mv 0 rSin 0 mvr 2 gr v0 Cos 0 2. For a rigid body (rotating about a fixed axis) Demo: the gyroscope compass the helicopter For a nonrigid body (rotating about a fixed axis) • Can change rotational inertia through internal forces Examples: rotating skater, collapsing interstellar dust cloud • Angular momentum must remain constant • Decreased rotational inertia means increased angular speed, and vice versa video Act A student sits on a rotating stool with his arms extended and a weight in each hand. He then pulls his hands in toward his body. In doing this her kinetic energy (a) increases (b) decreases (c) stays the same f i Ii If 3. For system of particle and rigid body If there is no net external torque on a system, angular momentum is conserved Example A uniform stick of mass M and length D is pivoted at the center. A bullet of mass m is shot through the stick at a point halfway between the pivot and the end. The initial speed of the bullet is v1, and the final speed is v2. – What is the angular speed F of the stick after the collision? M m D D/4 F v1 v2 initial final Solution M m D D/4 F v1 v2 initial final Conserve angular momentum around the pivot (z) axis! Act A uniform stick of mass m and length l is moving on a frictionless table with velocity v. It hits the nail at a distance l/4 from one end, so it rotates about the nail. Find the angular velocity. Solution Conservation of angular momentum r 8-5 Precession Consider a top that is supported at one point O. If the top is spinning in a very high speed, one possible motion is a steady circular motion of the axis combined with the spin motion of the top about the symmetry axis We call this phenomenon precession The angular momentum of top The angular momentum of top changes only its direction, but not its magnitude. rc mg O • A torque perpendicular to the axis of rotation can cause the axis itself to rotate • If there is no such torque, the axis will not rotate – this leads to the stability of gyroscopes Find the angular velocity of Precession d The Dynamics of Rotation the angular frequency of precession Discussion O 1. the angular frequency of precession depends on , is independent of 2. the direction of angular velocity of Precession depends on the direction of 3. There is another type of motion called nutation when is not large. CAI • Suppose you have a spinning top (gyroscope) in the configuration shown below. If the left support is removed, what will happen? – The gyroscope does not fall down! Instead it precesses around its pivot axis ! support o pivot d o pivot mg d top view Demo: the gyroscope in the form of a wheel pivot Summary of Chapter 8 • Angular momentum can be defined for any moving object; for an object moving at constant velocity, it is constant •Rotational quantities are analogous to linear quantities • Angular momentum is conserved in the absence of external torques Summary of Chapter 8, cont. • Nonrigid objects can change rotational inertia through internal forces • Angular momentum, Iω, remains constant • The axis of rotation can itself experience a torque, which tends to change its direction