This is my effort personally.So, I want to share my idea and knowledge.

1. UNIVERSITY OF
GUJRAT,(Gujrat,Pakistan )

3. 
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·


4.  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.

5.   
 
L  r x p  m  r x v 

6.  For linear motion, the change in total linear
momentum of a system is related to the net external

force by

Δp
Fnet 
Δt
 For rotational motion, the change in angular
momentum of a system is related to the net torque by
 For net Torque
 ne t
 ne t
I  Δω ΔI  ω  ΔL
 I  


Δt
Δt
Δt
ΔL

Δt

7.  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

8.  Angular momentum of a system of particles is
 Rate of change of a vector

 

L   mi (ri  ri )
i


 
 )  (r )  ω  r

(ri s
i r
i
 For a rigid body, in the rotating frame of reference, all
the distances between the points of the rigid body are
fixed:


(ri )r  0

 
 )  ω r
(ri s
i

9.  Angular momentum of rigid body become

  
L   mi (ri  (ω  ri ))
i
 3
 3

L j   mi    jkl rik    lmn ωm rin  


 k ,l 1
i
 m,n 1



i
3

k ,l , m , n 1
 r r ωm mi
jkl lmn ik in

10. Angular momentum of a rigid body
Lj  
i

i
3

k ,l , m , n 1
3
 (
k , m , n 1
jm
 r r ωm mi
jkl lmn ik in
 kn   jn km )rik rin ωm mi
3
3
  ωk  mi [( ri ) 2  jk  rij rik ]   I jk ωk
k 1
k 1
i
L  Iω
• Rotational kinetic energy:
~
~ Iω ωL
~
Lω
ω

TR 

2
2
2

11.  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

12.  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.


13.  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.

14. 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 gyrocompass.
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

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. 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.

18. 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.

19.  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

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