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 (  ) --  = 2  f = 2  /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) = x m cos(  t +  )  Amplitude (x m ) --  Phase angle (  ) --  Remember that (  t+  ) is in radians

7. SHM Formula Reference

8. SHM in Action  Consider SHM with  =0: x = x m cos(  t)   t=0,  t=0, cos (0) = 1   t=1/2T,  t= , cos (  ) = -1   t=T,  t=2 , cos (2  ) = 1 

9. SHM Monster RestMinMax 10m

10. Phase   The value of  relative to 2  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)=-  x m sin(  t +  )   Since the particle moves from +x m to -x m 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)=-   x m cos(  t +  )   Making a substitution yields: a(t)=-  2 x(t)

14. SHM Monster RestMinMax 10m

15. Displacement, Velocity and Acceleration  Consider SMH with  =0: x = x m cos(  t) v = -  x m sin(  t) a = -   x m cos(  t)   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=-  2 x  But, F=ma so,  Since m and  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/2  ) 2