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Chapter 11 Lecture 19: Angular Momentum: II HW7 (problems): 9.61, 9.75, 10.2, 10.13, 10.30, 10.47, 10.51, Due on Friday, Mar. 27.

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Moments of Inertia of Various Rigid Objects

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11.8: Newton’s 2 nd Law in Angular Form The (vector) sum of all the torques acting on a particle is equal to the time rate of change of the angular momentum of that particle.

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Sample problem: Torque, Penguin Fall Calculations: The magnitude of l can be found by using The perpendicular distance between O and an extension of vector p is the given distance D. The speed of an object that has fallen from rest for a time t is v =gt. Therefore, To find the direction of we use the right-hand rule for the vector product, and find that the direction is into the plane of the figure. The vector changes with time in magnitude only; its direction remains unchanged. (b) About the origin O, what is the torque on the penguin due to the gravitational force ? Calculations: Using the right-hand rule for the vector product we find that the direction of t is the negative direction of the z axis, the same as l.

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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 The angular momentum of an individual particle is L i = m i r i 2 and are directed along the z axis

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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 This also gives the rotational form of Newton’s Second Law

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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 axis If 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

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Angular Momentum of a Bowling Ball The momentum of inertia of the ball is 2/5MR 2 The angular momentum of the ball is L z = I The direction of the angular momentum is in the positive z direction

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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 Net torque = 0 -> means that the system is isolated For a system of particles,

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Conservation of Angular Momentum, cont If the mass of an isolated system undergoes redistribution, the moment of inertia changes The conservation of angular momentum requires a compensating change in the angular velocity I i i = I f f = constant 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

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Conservation Law Summary For an isolated system - (1) Conservation of Energy: E i = E f (2) Conservation of Linear Momentum: (3) Conservation of Angular Momentum: (demo)

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Joe has volunteered to help out in his physics class by sitting on a stool that easily rotates. As Joe holds the dumbbells out as shown, the professor temporarily applies a sufficient torque that causes him to rotate slowly. Then, Joe brings the dumbbells close to his body and he rotates faster. Why does his speed increase? a) By bringing the dumbbells inward, Joe exerts a torque on the stool. b) By bringing the dumbbells inward, Joe decreases the moment of inertia. c) By bringing the dumbbells inward, Joe increases the angular momentum. d) By bringing the dumbbells inward, Joe increases the moment of inertia. e) By bringing the dumbbells inward, Joe decreases the angular momentum.

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Joe has volunteered to help out in his physics class by sitting on a stool that easily rotates. As Joe holds the dumbbells out as shown, the professor temporarily applies a sufficient torque that causes him to rotate slowly. Then, Joe brings the dumbbells close to his body and he rotates faster. Why does his speed increase? a) By bringing the dumbbells inward, Joe exerts a torque on the stool. b) By bringing the dumbbells inward, Joe decreases the moment of inertia. c) By bringing the dumbbells inward, Joe increases the angular momentum. d) By bringing the dumbbells inward, Joe increases the moment of inertia. e) By bringing the dumbbells inward, Joe decreases the angular momentum.

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Joe has volunteered to help out in his physics class by sitting on a stool that easily rotates. Joe holds the dumbbells out as shown as the stool rotates. Then, Joe drops both dumbbells. How does the rotational speed of stool change, if at all? a) The rotational speed increases. b) The rotational speed decreases, but Joe continues to rotate. c) The rotational speed remains the same. d) The rotational speed quickly decreases to zero rad/s.

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Joe has volunteered to help out in his physics class by sitting on a stool that easily rotates. Joe holds the dumbbells out as shown as the stool rotates. Then, Joe drops both dumbbells. How does the rotational speed of stool change, if at all? a) The rotational speed increases. b) The rotational speed decreases, but Joe continues to rotate. c) The rotational speed remains the same. d) The rotational speed quickly decreases to zero rad/s.

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11.11: Conservation of Angular Momentum If the component of the net external torque on a system along a certain axis is zero, then the component of the angular momentum of the system along that axis cannot change, no matter what changes take place within the system. (a)The student has a relatively large rotational inertia about the rotation axis and a relatively small angular speed. (b)By decreasing his rotational inertia, the student automatically increases his angular speed. The angular momentum of the rotating system remains unchanged.

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Sample problem Figure a shows a student, sitting on a stool that can rotate freely about a vertical axis. The student, initially at rest, is holding a bicycle wheel whose rim is loaded with lead and whose rotational inertia I wh about its central axis is 1.2 kg m 2. (The rim contains lead in order to make the value of I wh substantial.) The wheel is rotating at an angular speed wh of 3.9 rev/s; as seen from overhead, the rotation is counterclockwise. The axis of the wheel is vertical, and the angular momentum L wh of the wheel points vertically upward. The student now inverts the wheel (Fig b) so L wh that, as seen from overhead, it is rotating clockwise. Its angular momentum is now - L wh.The inversion results in the student, the stool, and the wheel’s center rotating together as a composite rigid body about the stool’s rotation axis, with rotational inertia I b = 6.8 kg m 2. With what angular speed w b and in what direction does the composite body rotate after the inversion of the wheel?

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Sample problem

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Motion of a Top The only external forces acting on the top are the normal force and the gravitational force The direction of the angular momentum is along the axis of symmetry The right-hand rule indicates that the torque is in the xy plane

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Motion of a Top, cont The net torque and the angular momentum are related: A non-zero torque produces a change in the angular momentum The result of the change in angular momentum is a precession about the z axis The direction of the angular momentum is changing The precessional motion is the motion of the symmetry axis about the vertical The precession is usually slow relative to the spinning motion of the top

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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 axle The normal force produces no torque

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11.12: Precession of a Gyroscope (b) A rapidly spinning gyroscope, with angular momentum, L, precesses around the z axis. Its precessional motion is in the xy plane. (c) The change in angular momentum, dL/dt, leads to a rotation of L about O. (a) A non-spinning gyroscope falls by rotating in an xz plane because of torque .

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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 the total momentum of the system remains zero

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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:

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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 gives I i i = I f f = constant

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