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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
Simple Harmonic Motion

Physics 202

Professor Lee Carkner

Lecture 3


Pal 2 archimedes
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


Simple harmonic motion1
Simple Harmonic Motion

  • A particle that moves between 2 extremes in a fixed period of time

  • Examples:

    • mass on a spring

    • pendulum



  • Key quantities
    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


    Equation of motion
    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



    Shm in action
    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


    Shm monster
    SHM Monster

    Min

    Rest

    Max

    10m


    Phase
    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



    Velocity
    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


    Acceleration
    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)


    Shm monster1
    SHM Monster

    Min

    Rest

    Max

    10m


    Displacement velocity and acceleration
    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



    Force
    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?


    Linear oscillator
    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



    Application of the linear oscillator mass in free fall
    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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