Angular Momentum

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Chapter 10
Angular Momentum
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Angular Momentum
For a single particle,
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Torque and Angular Momentum (single particle)
The rate of change of angular momentum of a
particle is the net torque acting on it.
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Angular Momentum of a System of Particles
For a system, total angular momentum is the
vector sum of angular momenta of the individual
particles
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However,
The rate of change of angular momentum of a
system of particles is equal to the net external
torque on the system.
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Angular Momentum and Angular Velocity (For Fixed
Axis Rotation)
Consider a single particle of the system
However,
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Summing over all the particles of the body,
Conclusion The net angular momentum is, in
general, not equal to
The component of parallel to the axis of
rotation is rotational inertia about the axis
times angular velocity.
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Since Lz I?,
However,
(In general)
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For bodies symmetrical about the axis of
rotation, only the component along the axis of
rotation survives
O
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Rectangular plate will not keep rotating about
the vertical, whereas the square plate will.
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Angular Momentum of Rotating Rectangular Plate
dxdy
x
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Integrating over the entire plate,
Clearly, is not parallel to
However, if a b,
For a rectangular plate, however,
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Ex. 10.15
Cylinder 1 is rotating and 2 is stationary as
shown. When brought in contact, the two rotate
without slipping. What is the final angular speed
of 1.
Ans
If the angular impulses on the two spheres due to
the frictional forces are N1 N2, then
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Moreover, as slipping stops,
Solving for
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Conservation of Angular Momentum
In the absence of external torque, the total
angular momentum of a system is conserved
For angular momentum to be conserved, Newtons
third law must hold in its strong form.
Third Law in Week Form
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Newtons Third Law in Strong Form
each must be along the line joining the two
particles
For mechanical linear momentum to be conserved,
Newtons third law need hold only in the week
form. But for conservation of mechanical angular
momentum, it must hold in its strong form
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Example 1
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Example 2
Since PV of the planet is always perpendicular to
Ang. Momentum vector, the orbit of the planet is
confined to a plane
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Example 3
Can Two-body collision be 3 dimensional ?
Since is conserved and is
perpendicular to the plane of incidence and
is perpendicular to the scattering plane, the two
planes must be one and the same.
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Prob. 10.4
The cylinder is initially at rest. The block of
mass M moves over the cylinder from left to
right. If slipping stops before the block looses
contact with cylinder, find the final velocity of
block.
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System Cylinder Block
Angular momentum is conserved about the centre of
the cylinder.
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Prob. 10.8
A uniform flat disk of mass M and radius R
rotates about a horizontal axis with angular
speed ?0. A chip of mass m breaks off the edge at
an instant such that it rises vertically above
the point at which it broke off. A) How high
above the point does it rise ? B) What is the
final angular speed of the broken disk ?
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Ans
A) v R?0
B) The angular speed of the broken disk will
remain unchanged, as the broken chip simply
breaks off, carrying the same angular momentum as
it carried as part of the disk.
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Ex. 10.6
Prove that for a system of particles,
Where,
Ang. Momentum about an arbitrry point O
Ang. Momentum about CM
Position vector of CM w.r.t. O
Velocity of CM w.r.t. O
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Differentiating w.r.t. time
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The last two terms are zero, as, are both
zero. Hence the result.
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Orbital and Spin Angular Momenta of Earth
is conserved.
It can however be shown that the two angular
momenta are separately conserved.
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Precession of the Top
Because of the equatorial bulge of the earth, the
net torque on the earth is not zero.
This causes the earths axis to precess around
the normal to its orbital plane.
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It takes 26,000 years for the axis to go round
the vertical once.
Consequence 13,000 years from now, it will be
summer in December and winter in June