1. Angular Momentum Dual Credit Physics Montwood High School R. Casao

2. Rotating objects have angular momentum L. Angular momentum is the rotational equivalent of linear momentum. Changes in angular momentum are associated with torques. –Torque is the product of a moment arm and a force. –Angular momentum is the product of a moment arm and a linear momentum. –For a particle of mass m, the linear momentum is p = m·v, where v = r· . –The angular momentum is: L = r  ∙p = m∙ r  ∙v = m∙ r  ∙ r  ·  = m∙r  2 ·  –Unit:

3. For circular motion, r  = r, since velocity is perpendicular to r. For a system of particles making up a rigid body, all the particles travel in circles and the magnitude of the total angular momentum is: L = (  m i ∙r i 2 )∙  = I∙  For rotation about a fixed axis (in vector notation): Angular momentum L is in the direction of the angular velocity vector . The direction is given by the right hand rule.

4. The figure shows a particle of mass m with linear momentum p = m·v as it passes thru point a in the xy plane. The angular momentum L of the particle with respect to the origin O is: r is the position vector of the particle wrt O. As the particle moves wrt O in the direction of its momentum p, the position vector r rotates around O.

5. To have angular momentum, the particle itself does not have to rotate around O. The direction of the angular momentum vector is found by sliding the vector p until its tail is at the origin O. Use the right-hand rule to cross vector r into vector p: –fingers of the right hand point in the direction of r.

6. –Curl the fingers in the direction of vector p. –The thumb points in the direction of the angular momentum vector L. The magnitude of the angular momentum vector is: L = r·m·v·sin , where  is the angle between r and p when the two vectors are tail to tail. Angular momentum only has meaning wrt a specific origin. –The direction of the angular momentum vector is always perpendicular to the plane formed by the position vector and the linear momentum vector.

7. For linear motion, the change in total linear momentum of a system is related to the net external force by: For rotational motion, the change in angular momentum of a system is related to the net torque by: The net torque is equal to the time rate of change of angular momentum. A net torque causes a change in angular momentum.

8. If the net external torque acting on a system is zero, the angular momentum L of the system remains constant, no matter what changes take place within the system: L i = L f ;  I i ·  i =  I f ·  f Because momentum is a vector quantity, the conservation of momentum has to be considered in all three dimensions (x, y, z). Depending on the torques acting on a system, the angular momentum of the system might be conserved in only 1 or 2 directions but not in all directions. Conservation of Angular Momentum

9. For a rigid body with a constant moment of inertia I, the angular speed  remains constant in the absence of a net torque. It is possible for the moment of inertia to change in some systems, giving rise to a change in the angular speed. When the string is pulled downward through the tube, the revolving ball speeds up.

10. Conservation of angular momentum explains Kepler’s law of equal areas. –A planet’s angular momentum is conserved if we neglect the weak gravitational torques from other planets. –When a planet is closer to the Sun in its elliptical orbit, the moment arm is shorter, therefore, its speed is greater by the conservation of angular momentum. Conservation of angular momentum explains the large wind speeds of tornadoes and hurricanes. –As air rushes in toward the center of the storm where the pressure is low, its rotational speed must increase for angular momentum to be conserved. –Angular momentum L constant, as rotational inertia I becomes smaller as radius r decreases, angular velocity  increases.

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12. Angular momentum L is a vector and when it is conserved or constant, its magnitude and direction must remain unchanged. –When no external torques act, the direction of L is fixed in space. –This is the principle behind passing a football accurately and providing directional stability for bullets fired from guns. –The spiraling rotation stabilizes the spin axis in the direction of motion.

13. Gyroscopes work in similar fashion. –Once you spin a gyroscope (the L vector is set in a particular direction), its axle wants to keep pointing in the same direction. If you mount the gyroscope in a set of gimbals so that it can continue pointing in the same direction, it will. This is the basis of the gyro-compass. –In the absence of external torques, the gyroscope direction remains fixed, even though the carrier (plane or ship) changes directions. When the frame moves, the gyroscope wheel maintains its direction. –The gyroscope’s center of gravity is on the axis of rotation, so there is no net torque due to its weight.

14. –If you mount two gyroscopes with their axles at right angles to one another on a platform, and place the platform inside a set of gimbals, the platform will remain completely rigid as the gimbals rotate in any way they please. This is this basis of inertial navigation systems (INS). –In an INS, sensors on the gimbals' axles detect when the platform rotates. The INS uses those signals to understand the vehicle's rotations relative to the platform. If you add to the platform a set of three sensitive accelerometers, you can tell exactly where the vehicle is heading and how its motion is changing in all three directions. With this information, an airplane's autopilot can keep the plane on course, and a rocket's guidance system can insert the rocket into a desired orbit!

15. –With this information, an airplane's autopilot can keep the plane on course, and a rocket's guidance system can insert the rocket into a desired orbit. –The gyroscope eventually slows down due to friction. This causes the L vector to tilt. –The spin axis revolves or precesses about the vertical axis because the L vector is no longer constant in direction, indicating that a torque must be acting to produce the change  L with time. –The torque arises from the vertical component of the weight, which no longer lies above the point of support or on the vertical axis of rotation. –The instantaneous torque causes the gyroscope’s axis to move or precess about the vertical axis.

16. –The Earth’s rotational axis also precesses. –The spin axis is tilted 23.1º wrt a line perpendicular to the plane of its revolution around the Sun. –The Earth’s axis precesses about this line vertical line due to the gravitational torques exerted on the Earth by the Moon and the Sun. –The precession occurs over a period of 26000 years and causes the north star to change over long periods of time.

17. The Earth’s daily spin rate is slowing down and the days are getting longer. –This is due to the torque acting on the Earth due to ocean tidal friction. The Earth’s spin angular momentum is slowing down and the average day will be about 25 s longer this century as a result. The tidal torque acting on the Earth results from the Moon’s gravitational attraction. –This torque is internal to the Earth-Moon system, so the total angular momentum of the system is conserved. –Since the Earth is losing angular momentum, the Moon must be gaining angular momentum to keep the total angular momentum constant. –As a result the Moon drifts slightly farther from the Earth (about 4 cm per year) and its orbital speed decreases.

18. The Earth’s rotation speeds up for short periods of time as a result of changes associated with the rotational inertia of the liquid layer of the Earth’s core. Angular Momentum and Helicopters the motor supplying the torque to rotate the front rotor is internal and angular momentum must be conserved. –Initially, L = 0; to conserve the total angular momentum of the rotor-body system, the angular momentum of the rotor and the helicopter body would have to be in opposite directions to cancel.

19. –On takeoff, the rotor would rotate one way and the helicopter body would rotate in the opposite direction. To prevent this situation, helicoptors have two rotors.

20. –Large helicopters have two overlapping rotors. The oppositely rotating rotors cancel each other’s angular momenta so the helicopter body does not have to rotate to provide cancellation of the rotor’s angular momentum. The two rotors are offset at different heights so they do not collide with each other. –Small helicopters with a single overhead rotor have small “antitorque” tail rotors. The tail rotor produces a thrust like a propeller and supplies the torque to counterbalance the torque produced by the overhead rotor. The tail rotor also helps steer the helicopter. Increasing or decreasing the tail rotor’s thrust causes the helicopter to rotate one way or the other.