Simple harmonic motion
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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

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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


  • Shm snapshots

    SHM Snapshots


    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 formula reference

    SHM Formula Reference


    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


    Amplitude period and phase

    Amplitude, Period and Phase


    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


    Derivatives of shm equation

    Derivatives of SHM Equation


    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


    Linear oscillator1

    Linear Oscillator


    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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