Conservation of Angular Momentum

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Conservation of Angular Momentum

• Definitions

Since
Consider
Note
Thus
Conclusion
The angular momentum of particle subject to no
torque is conserved.
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Work

• Definition (Just a reminder)

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1
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Kinetic Energy

• To motivate the concept, consider

The work, W, can thus be expressed as an exact
differential.
Definition of Kinetic Energy
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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

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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.
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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.
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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.

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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.
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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!
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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!

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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.

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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).

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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)...
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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
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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
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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
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Stable/Unstable equilibrium (contd)

• We have an equilibrium if

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