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Understanding Simple Harmonic Motion in Physics 215

Explore oscillations, restoring forces, differential equations, harmonic oscillators, periodic motion, resonance, and oscillatory phenomena in physical systems. Learn about pendulums, springs, and Newton's 2nd law for blocks in this comprehensive study of simple harmonic motion.

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Understanding Simple Harmonic Motion in Physics 215

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  1. Welcome back to Physics 215 • Oscillations • Simple harmonic motion

  2. Current homework assignment • HW10: • Knight Textbook Ch.14: 32, 52, 56, 74, 76, 80 • Due Wednesday, Nov. 19th in recitation

  3. Oscillations • Restoring force leads to oscillations about stable equilibrium point • Consider a mass on a spring, or a pendulum • Oscillatory phenomena also in many other physical systems...

  4. F=0 Spring constant x 0 F F F Simple Harmonic Oscillator Fx= -kx Newton’s 2nd Law for the block: Differential equation for x(t)

  5. Simple Harmonic Oscillator Differential equation for x(t): Solution:

  6. Simple Harmonic Oscillator initial phase angular frequency amplitude Units: A - m T - s f - 1/s = Hz (Hertz) w - rad/s f – frequency Number of oscillations per unit time T – Period Time taken by one full oscillation

  7. Simple Harmonic OscillatorDEMO • stronger spring (larger k) a faster oscillations (larger f) • larger mass a slower oscillations

  8. Simple Harmonic OscillatorTotal Energy E =

  9. Simple Harmonic Oscillator -- Summary If F= -kx then

  10. Importance of Simple Harmonic Oscillations • For all systems near stable equilibrium • Fnet ~ - x where x is a measure of small deviations from the equilibrium • All systems exhibit harmonic oscillations near the stable equilibria for small deviations • Any oscillation can be represented as superposition (sum) of simple harmonic oscillations (via Fourier transformation) • Many non-mechanical systems exhibit harmonic oscillations (e.g., electronics)

  11. L T m 0 x Fnet q mg (Gravitational) Pendulum Simple Pendulum – Point-like Object Fnet = mg sinq q For smallq, Fnetis in –x direction: Fx = - mg/L x “Pointlike” – size of the object small compared to L DEMO

  12. Two pendula are created with the same length string. One pendulum has a bowling ball attached to the end, while the other has a billiard ball attached. The natural frequency of the billiard ball pendulum is: 1. greater 2. smaller 3. the same as the natural frequency of the bowling ball pendulum.

  13. The bowling ball and billiard ball pendula from the previous slide are now adjusted so that the length of the string on the billiard ball pendulum is shorter than that on the bowling ball pendulum. The natural frequency of the billiard ball pendulum is: 1. greater 2. smaller 3. the same as the natural frequency of the bowling ball pendulum.

  14. T d q m, I mg (Gravitational) Pendulum Physical Pendulum – Extended Object tnet = d mg sinq q For smallq : sinq ≈q tnet = d mg q DEMO

  15. A pendulum consists of a uniform disk with radius 10cm and mass 500g attached to a uniform rod with length 500mm and mass 270g. What is the period of its oscillations?

  16. Torsion constant Torsion Pendulum (Angular Simple Harmonic Oscillator) t=-kq Solution: DEMO

  17. Differential equation for x(t) Damping constant Damped Harmonic Oscillator For example air drag for small speeds: Fdrag= - b v Solution:

  18. Forced Harmonic Oscillator Differential equation for x(t): Driving force Damping force Natural frequency Solution yields amplitude of response:

  19. Small (b) damping Large (b) damping Resonance A d

  20. Reading assignment • Gravity • Chapter 13 in textbook

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