Simple Harmonic Motion

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 = mbowl2 • Volume of object = • Density of object = mobj / Vobj • Biggest source of error is getting the right amount of water into the cup • class answers range from 3000-15000 kg/m3 • 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 (w) -- w = 2pf = 2p/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) = xmcos(wt + f) • Amplitude (xm) -- • Phase angle (f) -- • Remember that (wt+f) is in radians

5. SHM in Action • Consider SHM with f=0: x = xmcos(wt) • Remember w=2p/T • x=xm • x=-xm • x=xm

6. SHM Monster Min Rest Max 10m

7. 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

8. 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

9. Acceleration a(t)=-w2xmcos(wt + f) • Making a substitution yields: a(t)=-w2x(t)

10. x, v and a • Consider SMH with f=0: x = xmcos(wt) v = -wxmsin(wt) = -vmsin(wt) a = -w2xmcos(wt) = -amcos(wt) • 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=-mw2x F=-kx • Where k=mw2 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=mw2

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, • Higher • Lower • 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, • Higher • Lower • 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, • Higher • Lower • 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, • Both the same • Both higher • Both lower • Flow rate is same, velocity is higher • 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, • Higher • Lower • The same • 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 • Higher • Lower • The same • Can’t tell