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

2. PAL #2 Archimedes  Weigh object =  Weigh bowl =  Put cup full of water in bowl, insert object, remove cup  Weigh bowl of water = m bowl2  Volume of object =  Density of object = m obj / V obj  Biggest source of error is getting the right amount of water into the cup   class answers range from 3000-15000 kg/m 3  Would not work for floating object unless you held it underwater  Hard to estimate fraction submerged

3. Simple Harmonic Motion   e.g. a mass on a spring  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

4. Equation of Motion   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

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

6. SHM Monster RestMinMax 10m

7. Phase   The value of  relative to 2  indicates the offset as a function of one period   It is phase shifted by 1/2 period

8. 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

9. Acceleration  a(t)=-   x m cos(  t +  )   Making a substitution yields: a(t)=-  2 x(t)

10. x, v and a  Consider SMH with  =0: x = x m cos(  t) v = -  x m sin(  t) = -v m sin(  t) a = -   x m cos(  t) = - a m cos(  t)   Mass is momentarily at rest, but being pulled hard in the other direction   Mass coasts through the middle at high speed

11. Force   But, F=ma so, F=-m  2 x  F=-kx  Where k=m  2 is the spring constant    Simple harmonic motion is motion where force is proportional to displacement but opposite in sign  Why is the sign negative?

12. Linear Oscillator  A simple 1-dimensional SHM system is called a linear oscillator   In such a system, k=m  2 

13. Next Time  Read: 15.4, 15.6, 15.8, 15.9  Homework: Ch 15, P: 9, 28, 46, 78, 109

14. Consider a small ball on a toy ship floating in a tank of water. If you remove the ball from the ship and place it on the table the water level will be, a)Higher b)Lower c)The same

15. Consider a small ball on a toy ship floating in a tank of water. If you remove the ball from the ship and place it in the tank and it floats, the water level will be, a)Higher b)Lower c)The same

16. Consider a small ball on a toy ship floating in a tank of water. If you remove the ball from the ship and place it in the tank and it sinks to the bottom, the water level will be, a)Higher b)Lower c)The same

17. Water flows through a horizontal pipe from point A to point B. If the pipe is narrower at B than at A the flow rate and velocity at point B compared to point A are, a)Both the same b)Both higher c)Both lower d)Flow rate is same, velocity is higher e)Velocity is same, flow rate is higher

18. Water flows uphill through a pipe from point A to point B. If the velocity is the same at points A and B, the pressure at B compared to A is, a)Higher b)Lower c)The same d)Can’t tell

19. Water flows through a horizontal pipe from point A to point B. If the pressure at point B is higher than that at point A, the velocity at point B compared to A is a)Higher b)Lower c)The same d)Can’t tell