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Conservation of Angular Momentum

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Title: PowerPoint Presentation Author: Claude Pruneau Last modified by: Claude Pruneau Created Date: 1/2/2006 6:39:53 PM Document presentation format – PowerPoint PPT presentation

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Title: Conservation of Angular Momentum


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(No Transcript)
2
Conservation of Angular Momentum
  • Definitions

Since
Consider
Note
Thus
Conclusion
The angular momentum of particle subject to no
torque is conserved.
3
Work
  • Definition (Just a reminder)

2
1
4
Kinetic Energy
  • To motivate the concept, consider

The work, W, can thus be expressed as an exact
differential.
Definition of Kinetic Energy
5
Conservative Forces
If the work performed by a force while moving a
particle between two given (arbitrary) positions
is independent of the path followed, then the
work can be expressed as a function of the two
end points of the path.
Conservative Force or system
  • Where we defined the functions Ui as the
    potential energy of the particle at the location
    i.
  • Note the signs

6
Conservative Forces (contd)
  • A force is conservative if it can be expressed a
    the gradient of a scalar function U.

Verify by substitution
In most systems of interest,U is a function of
the position only, or position and time. We will
study central potentials in particular, and we
will not consider potentials that depend on
velocity.
7
Conservative Forces (contd)
  • Important notes about potentials.
  • Potentials are defined only up to a constant
    since

Potentials are known relative to a chosen
(arbitrary) reference. Choose reference position
and values to ease the solution of specific
problems. E.g. for 1/r potentials, choose U0 at
infinity.
8
Conservative Forces (contd)
  • Potential energy is thus NOT an absolute
    quantity it does not have an absolute value.
  • Likewise, the Kinetic Energy is also NOT an
    absolute quantity it depends on the specific
    rest frame used to measure the velocity.

9
Total Mechanical Energy
  • Definition E T U.
  • It is a conserved quantity!
  • To verify, consider the time derivative

Recall
Thus
The time derivative of the potential can be
expressed as a sum of partial derivatives.
10
Total Mechanical Energy (contd)
  • So adding the 2 terms

0
Conclusion if U is not an explicit function of
time, then the energy is conserved!
11
Total Mechanical Energy (contd)
  • In a conservative system, the force can be
    expressed as a function of a gradient of a
    potential independent of time.
  • The total mechanical energy, E, is thus a
    conserved quantity in a conservative system.
  • The conservations theorem we just saw can be
    considered as laws, but keep in mind they
    strictly equivalent to Newtons Eqs 2 3.
  • Conservations theorems are elegant, and
    powerful.
  • They led W. Pauli (1880-1958) to postulate (in
    1930) the existence of the neutrino, as a product
    of b-decay to explain the observed missing
    momentum!

12
Example Mouse on a fan
  • Question A mouse of mass m jumps on the
    outside of a freely spinning ceiling fan of
    moment of inertia I and radius R. By what ratio
    does the angular velocity change?
  • Answer
  • Angular momentum must be conserved.
  • Calculate the angular momentum before and after
    the jump.
  • Equate them.

13
Energy
  • Concept of energy now more popular than in
    Newtons time
  • Became clear early 19th century that other forms
    of energy exist e.g. heat.
  • Rutherford discovered clear link between heat
    generation and friction.
  • Law of conservation of energy first formulated by
    Hermann von Helmholtz (1821-1894) based on
    experimental work done largely by James Prescott
    Joule (1818-1889).

14
Use of Energy for problem solving.
  • Total mechanical energy

1-D Case
This is a generic solution need U(x) and
integrate to get a function of t(x)...
15
Energy (contd)
  • Can learn a great deal without performing the
    integration (which can get difficult).
  • Consider a plot of the energy and potential vs x.

EE4 - unbound motion
EE3 - 1 side bound, non periodic
EE2 - bound periodic motion
EE1 - bound periodic motion
16
Energy (contd)
  • Note Whenever motion is restricted near a
    minimum of a potential, it may be sufficient to
    approximate U(x) with a harmonic potential
    approximation

U(x)
E
17
Stable/Unstable equilibrium
  • One can determine whether an equilibrium is
    stable or unstable base on the curvature of the
    potential at the equilibrium point.

Consider a Taylor expension of the potential
18
Stable/Unstable equilibrium (contd)
  • We have an equilibrium if

Near xo
Stable equilibrium if
Unstable equilibrium if
Higher orders to be considered if
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