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Simple Harmonic Motion

Simple Harmonic Motion. Physics 202 Professor Lee Carkner Lecture 3. PAL #2 Archimedes. a) Iron ball removed from boat Boat is lighter and so displaces less water b) Iron ball thrown overboard While sinking iron ball displaced water equal to its volume c) Cork ball thrown overboard

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Simple Harmonic Motion

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  1. Simple Harmonic Motion Physics 202 Professor Lee Carkner Lecture 3

  2. PAL #2 Archimedes • a) Iron ball removed from boat • Boat is lighter and so displaces less water • b) Iron ball thrown overboard • While sinking iron ball displaced water equal to its volume • c) Cork ball thrown overboard • Both ball and boat still floating and so displaced amount of water is the same

  3. Simple Harmonic Motion • A particle that moves between 2 extremes in a fixed period of time • Examples: • mass on a spring • pendulum

  4. SHM Snapshots

  5. Key Quantities • Frequency (f) -- • Unit=hertz (Hz) = 1 oscillation per second = s-1 • Period (T) -- • T=1/f • Angular frequency (w) -- w = 2pf = 2p/T • Unit = • We use angular frequency because the motion cycles

  6. Equation of Motion • What is the position (x) of the mass at time (t)? • The displacement from the origin of a particle undergoing simple harmonic motion is: x(t) = xmcos(wt + f) • Amplitude (xm) -- • Phase angle (f) -- • Remember that (wt+f) is in radians

  7. SHM Formula Reference

  8. SHM in Action • Consider SHM with f=0: x = xmcos(wt) • t=0, wt=0, cos (0) = 1 • t=1/2T, wt=p, cos (p) = -1 • t=T, wt=2p, cos (2p) = 1

  9. SHM Monster Min Rest Max 10m

  10. Phase • The value of f relative to 2p indicates the offset as a function of one period • It is phase shifted by 1/2 period

  11. Amplitude, Period and Phase

  12. Velocity • If we differentiate the equation for displacement w.r.t. time, we get velocity: v(t)=-wxmsin(wt + f) • Since the particle moves from +xm to -xm the velocity must be negative (and then positive in the other direction) • High frequency (many cycles per second) means larger velocity

  13. Acceleration • If we differentiate the equation for velocity w.r.t. time, we get acceleration a(t)=-w2xmcos(wt + f) • Making a substitution yields: a(t)=-w2x(t)

  14. SHM Monster Min Rest Max 10m

  15. Displacement, Velocity and Acceleration • Consider SMH with f=0: x = xmcos(wt) v = -wxmsin(wt) a = -w2xmcos(wt) • Mass is momentarily at rest, but being pulled hard in the other direction • Mass coasts through the middle at high speed

  16. Derivatives of SHM Equation

  17. Force • Remember that: a=-w2x • But, F=ma so, • Since m and w are constant we can write the expression for force as: F=-kx • This is Hooke’s Law • Simple harmonic motion is motion where force is proportional to displacement but opposite in sign • Why is the sign negative?

  18. Linear Oscillator • Example: a mass on a spring • We can thus find the angular frequency and the period as a function of m and k

  19. Linear Oscillator

  20. Application of the Linear Oscillator: Mass in Free Fall • However, for a linear oscillator the mass depends only on the period and the spring constant: m/k=(T/2p)2

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