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 Conservative Force or system
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 (cont’d)
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 (cont’d)
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 U=0 at infinity.

8. Conservative Forces (cont’d)
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 (cont’d)
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 (cont’d)
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 Newton’s Eqs 2 & 3.
Conservations theorems are elegant, and powerful.
They led W. Pauli ( ) 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 Newton’s 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 ( ) based on experimental work done largely by James Prescott Joule ( ).

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 (cont’d)
Can learn a great deal without performing the integration (which can get difficult…).
Consider a plot of the energy and potential vs x. E0 E1 E2 E3 E4 x
E=E4 - unbound motion
E=E3 - 1 side bound, non periodic
E=E2 - bound periodic motion
E=E1 - bound periodic motion

16. Energy (cont’d)
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.
stable
U(x) E
unstable
Consider a Taylor expension of the potential:

18. Stable/Unstable equilibrium (cont’d)
We have an equilibrium if:
Near xo:
Stable equilibrium if:
Unstable equilibrium if:
Higher orders to be considered if: