Oscillations about Equilibrium

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

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

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

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.

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

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

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:

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.

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.

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

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

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

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-1 7.7 = 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

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

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 340 + 0 (a) (b) 340 ─ 38 f’ = 1200 340 ─ 45 f’ = 1200 340 ─ 45 minus because source is moving towards the observer f’ = 1228.5 Hz f’ = 1383.1 Hz