1. Oscillations about Equilibrium

2. Simple Harmonic Motion
A spring exerts a restoring force that is proportional to the displacement from equilibrium:
F= kx

3. Describing vibrations
Amplitude - maximum extent of displacement from equilibrium
Cycle - one complete vibration
Period - time for one cycle
Frequency - number of cycles per second (units = hertz, Hz)
Period and frequency inversely related

4. Energy Conservation in Oscillatory Motion
In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring:
ETotal = KE + EPE
Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies: 2
KEmax = ½mvmax
EPEmax =½kA
@ equilibrium position 2
@ amplitude

5. Energy Conservation in Oscillatory Motion
This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.

6. Sound Intensity
The intensity of a sound is the amount of energy that passes through a given area in a given time.

7. Sound Intensity
Sound intensity from a point source will decrease as the square of the distance.

8. Sound Intensity
When you listen to a variety of sounds, a sound that seems twice as loud as another is ten times more intense. Therefore, we use a logarithmic scale to define intensity values.
Here, I0 is the faintest sound that can be heard:

9. Sound Intensity
The quantity β is called a bel; a more common unit is the decibel, dB, which is a tenth of a bel.
The intensity of a sound doubles with each increase in intensity level of 10 dB.

10. The Doppler Effect
The Doppler effect is the change in pitch of a sound when the source and observer are moving with respect to each other.
When an observer moves toward a source, the wave speed appears to be higher, and the frequency appears to be higher as well.

11. The Doppler Effect
The Doppler effect from a moving source can be analyzed similarly; now it is the wavelength that appears to change:

12. The Doppler Effect
Combining results gives us the case where both observer and source are moving:
f ’ =
v ± vo
v ± vs f ( )
Where: f ’ is the shifted frequency
f is the frequency of the source
v is the speed of sound in the medium (340 m/s in air)
vo is the speed of the observer
vs is the speed of the source
+ if observer is moving towards the source
top of the equation
- if observer is moving away from the source
bottom of the equation
+ if source is moving away from the observer
- if source is moving towards the observer

13. Sample Problem
If one cheerleader at a football stadium cheers at 77 dB, what is the intensity level of 18 cheerleaders each at 77 dB?
Solution:
Convert 77 dB to W/m2 I I
77 = 10 log
β = 10 log Io
1 x 10-12

14. I
77 = 10 log
Divide both sides by 10
1 x 10-12 I
Take the inverse log of both sides
7.7 = log
1 x 10-12 I I
log =
5.01 x 107 =
1 x 10-12
1 x 10-12
Solve for I
Use this
value as I
to determine
the
new β
I = 5.01 x 10-5 W/m2
Multiply by 18
5.01 x 10-5 x 18 = 9.02 x 10-4 W/m2

15. 9.02 x 10-4 I
β = 10 log
β = 10 log Io
1 x 10-12
β = 89.6 db

16. A police car at 45 m/s with its siren (f = 1200 Hz) blaring is chasing a car moving at 38 m/s. What frequency is heard by the (a) driver of the car being chased? (b) stationary gawkers as they approach?
f = 1200 Hz
Given:
vs = 45 m/s
vo= 38 m/s (driver); 0 m/s (gawkers)
zero because observers are stationary
f ’ =
v ± vo
v ± vs f ( )
v = 340 m/s (speed of sound)
minus because observer is moving away from the source
(a)
(b)
340 ─ 38
f’ = 1200
340 ─ 45
f’ = 1200
340 ─ 45
minus because source is moving towards the observer
f’ = Hz
f’ = Hz