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Conservation of Angular Momentum

Conservation of Angular Momentum. Definitions:. Since. Consider. Note. Thus. Conclusion:. The angular momentum of particle subject to no torque is conserved. Work. Definition (Just a reminder…). 2. 1. Kinetic Energy . To motivate the concept, consider:.

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Conservation of Angular Momentum

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  1. Conservation of Angular Momentum • Definitions: Since Consider Note Thus Conclusion: The angular momentum of particle subject to no torque is conserved.

  2. Work • Definition (Just a reminder…) 2 1

  3. Kinetic Energy • To motivate the concept, consider: The work, W, can thus be expressed as an exact differential. Definition of Kinetic Energy

  4. Conservative Forces • Where we defined the functions Ui as the potential energy of the particle at the location i. • Note the signs 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

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. 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!

  10. 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 (1880-1958) to postulate (in 1930) the existence of the neutrino, as a product of b-decay to explain the observed missing momentum!

  11. 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.

  12. 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 (1821-1894) based on experimental work done largely by James Prescott Joule (1818-1889).

  13. 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)...

  14. E4 E3 E2 E1 E0 x 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. E=E4 - unbound motion E=E3 - 1 side bound, non periodic E=E2 - bound periodic motion E=E1 - bound periodic motion

  15. 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

  16. unstable U(x) U(x) E E stable 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:

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

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