1 Formulae Sheet
Midterm Exam 2, Spring 2015 Conversion Factors:
1 m = 3.281ft; 1in = 2. 2.54 cm; cm; 1 mile mile = 5280 5280 ft; ft; 1 mile mile = 1. 1.61km; 3 ◦ 1 hr = 360 36000 s; 1 liter liter = 100 10000 cm ; 2π radians = 360 ; 1 N = 1 kgs·m ; 1Hz = 1 cycle . s 2
Standard Prefixes:
Giga Mega Mega kilo centi mill millii micro nano
109 106 103 10−2 10−3 µ 10−6 n 10−9
G M k c m
∆x xf − xi ; Average speed over time interval ∆ t: vave = ; Distance traveled: d = vt = vt.. ∆t tf − ti ∆v vf − vi m Acceleration: a = = ; Density: ρ = . ∆t tf − ti V Newton’s Second Law: F = ma; ma; Weight: W = mg (g = 9.8 m/s2 ). Velocity, speed: v =
Force due to a spring: F spring spring = −k ∆x. Pressure: P =
F ⊥ . A
1 2 Kinetic energy: KE = KE = 21 mv 2 ; Potential energy due to gravity: P E grav mgh , stored in spring: P E spring grav = mgh, spring = 2 kx . Conservation of energy: KE i + P E i = K = KE E f f + P E f f .
1 1 k 1 g Frequency, requency, period: f = ; Frequency mass/spring system: f = ; Frequency of pendulum: f = T 2π m 2π L v Wavelength: λ = . f Speed of Sound: F T T (i) On a string or wire: v = ; µ (ii) In air: v = (331. (331.3 + 0.6t) m/s, where t is temperature in ◦ C, i.e., at room temperature (t ( t = 20◦ C), vsound = 343..3 m/s. 343 Importance of diffraction: if encountering an opening of approximate diameter d, d , or an obstancle of typical size d size d,, diffraction is (i) important for λ ≥ d; (ii) much less important for λ d.
Interference due to path length difference:
1 (i) Constructive: |d2 − d1 | = nλ, nλ , n = 0, 1, 2, 3 . . . ; (ii) Destructive: |d2 − d1 | = n + 2
λ, n = n = 0, 1, 2, 3, . . . .
Change in frequency due to Doppler Effect: (i) Source moving, observer stationary: f obs obs
v = f = f vv s
sound
, (− approaching, + receding); ∓ vs sound ± vobs , (+ approaching, − approaching, − receding); vsound
sound
(ii) Observer moving, source stationary: f obs = f s obs = f
Beats: Beats: two two tones f A and f B presented simultaneously, beat frequency f beat beat = |f B − f A | is the frequency of the (f A + f B ) resultant resultant amplitude amplitude modulation modulation at a sum frequency frequency of f f sum . sum = 2 F T T Velocity of wave on a string or wire of tension force F T = m/L:: vwave = . T and mass per unit length µ = m/L µ
Harmonic Harmonic series series for transverse transverse standing waves waves on a string of length length L and mass per unit length µ = m/L = m/L:: nv n T 2L f n = nf = nf 1 = = , λ n = , where n where n = = 1, 2, 3, . . . 2L 2L µ n
2 Harmonic series for longitudinal standing waves in pipe/tube of length L with speed of sound v : 2L nv Open at both ends: λn = ; f n = = nf 1 , n = 1, 2, 3, · · · 2L n 4L nv Open at one end, closed on other: λn = ; f n = = nf 1 , n = 1, 3, 5, · · · (only odd values of n). n 4L Fourier Synthesis and Analysis: Any complex periodic wave with fundamental frequency f 1 can be built up using Fourier synthesis from an infinite harmonic series (i.e., f n = nf 1 ) of sinusoidal waves of different amplitudes, i.e., An sin(2πf n t). Some standard or well-known waves are: TABLE I: Fourier Amplitudes A1 = constant, all other A = 0
Waveform Sine
n
Sawtooth (Ramp)
A1 A = , n = 1, 2, 3, . . . n n
Square
A =
A1 , n = 1, 3, 5, . . . n
Triangle
A =
A1 , n = 1, 3, 5, . . . n2
n
n
Sound Levels: dB difference between two signals: dB = 10 log A A Sound Intensity Level (Li or S IL in dB): L i = 10log I I , relative to a reference sound intensity I 0 = 10−12 W/m2 . Sound Pressure Level (L p or S P L in dB): L p = 20 log pp , relative to a reference sound pressure p 0 = 2 × 10−5 Pa or N/m2 . In most normal situations, L p = L i . W Sound Power Level (in dB) of a source: LW = 10log W , relative to a reference sound power W 0 = 10−12 W. 1 2
0
0
0
x
Inverting a logarithm: if x = log(y), then y = 10 . If given two sources with sound intensity levels L I 1 and L I 2 in decibels, convert into two intensities I 1 and I 2 , add the intensities I tot = I 1 + I 2 , then convert I tot back to decibels. Definition of Intensity: I = W/A, where W is power, and A is area. W Variation of Intensity with Distance: I = where the source has sound power of W Watts and the intensity is 4πr 2 measured a distance of r meters away. Gain =
Output Quantity , Power Gain = 10 log(W 0 /W i ). Input Quantity
On an equal temperment scale (where one octave, i.e., a doubling or halving of frequency, is twelve semitones): to move up from a note of frequency f 0 by n octaves to frequency f : f 0 f = f 0 (2)n ; and to move down n semitones: f = n ; 2 to move up from a note of frequency f 0 by n semitones to frequency f : f 0 f = f 0 (1.0596)n; and to move down n semitones: f = . 1.0596n
