Angular momentum in terms of rigid body

UNIVERSITY OF GUJRAT,

(Gujrat,Pakistan)

Muhammad Zeeshan Afzal

Angular momentum in terms of Rigid body

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·

explanation

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 = (mi∙ri2)∙ = 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.

Cont….

vxrmpxrL

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

For net Torque

Cont….

Δt

pΔFnet

ΔtΔL

ΔtΔL

ΔtωIΔ

ΔtΔωI

I

net

net

a system of mass points subject to the holonomic constraints that the distances between all pairs of points remain constant throughout the motion

Rigid body has to be described by its orientation and location Position of the rigid body is determined by the position of any one point of the body, and the orientation is determined by the relative position of all other points of the body relative to that point

Rigid Body

Angular momentum of a system of particles is

Rate of change of a vector

For a rigid body, in the rotating frame of reference, all the distances between the points of the rigid body are fixed:

Angular Momentum of a Rigid Body

i

iii rrmL )(

irisi rωrr )()(

0)( rir

isi rωr )(

Angular momentum of rigid body become

Angular momentum of rigid body

i

iii rωrmL ))((

i lk nminmlmnikjklij rωrmL

3

1,

3

1,

i nmlk

iminiklmnjkl mωrr3

1,,,

i nmlk

iminiklmnjklj mωrrL3

1,,,

i nmk

iminikkmjnknjm mωrr3

1,,

)(

3

1

2 ])[(k

ikijjkii

ik rrrmω

3

1kkjkωI

IωL

Angular momentum of a rigid body

• Rotational kinetic energy:

2

~Lω

2

~ωL

2

~IωωRT

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

Applications

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.

continue

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.

Continue application

Application continueGyroscopes 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

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.

Application Cont….

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.

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. 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.

Cont….

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

Cont…

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 “anti torque” 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

Conti…

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