Springs & pendulums
- Mass on a spring
- T = 2π·√(m/k)
- Pendulum
- T = 2π·√(L/g)
- Angular frequency
- ω = √(k/m) = 2π/T
- Maximum speed and acceleration
- vmax = A·ω, amax = A·ω²
- Energy
- E = ½·k·A²
Waves and Optics: cram pack
Formula sheets, the key method steps for every topic and 4 practice problems, with full solutions at the back.
1Formulas and methods
Waves, sound & optics
- Wave speed
- v = fλ, T = 1/f
- String
- v = √(T/μ), μ = m/L
- Sound level
- β = 10·log10(I/I0), I0 = 10⁻¹² W/m²
- Point source
- I = P/(4πr²)
- Standing waves
- string or open pipe fn = n·v/(2L); closed pipe fn = n·v/(4L), n odd
- Doppler effect
- f′ = f·(v ± vo)/(v ∓ vs)
- Refractive index
- n = c/v, λ = λ0/n
- Snell’s law
- n1sin θ1 = n2sin θ2
- Critical angle
- sin θc = n2/n1
- Optical fibre
- NA = sin θa = √(n1² − n2²)
- Lenses and mirrors
- 1/f = 1/do + 1/di, m = −di/do, mirror f = R/2
- Double slit
- d·sin θ = mλ, ym = mλL/d
- Grating
- d = 1/N, d·sin θ = mλ, mmax ≤ d/λ
- Single slit
- a·sin θ = mλ (dark), w = 2λL/a
Method
- Identify the kind of problem: wave speed, sound level, standing wave, Doppler, refraction, lens or mirror, or interference.
- Write the relation first: v = fλ, β = 10·log(I/I₀), fₙ = n·v/(2L) or n·v/(4L), f′ = f(v ± vo)/(v ∓ vs), n₁sin θ₁ = n₂sin θ₂, 1/f = 1/d₀ + 1/dᵢ, d·sin θ = mλ.
- Convert units: nm, mm and cm to m, and check whether angles are measured from the normal.
- Check signs and plausibility: a virtual image has dᵢ < 0, an approaching source sounds higher, a closed pipe has only odd harmonics.
2Practice problems
1.Springs & pendulumsMedium
A 2.4 kg mass on a spring with k = 190 N/m is pulled 4 cm in the positive direction and released from rest at t = 0. Find its velocity v at t = 0.25 s (with sign).
m = 2.4 kgk = 190 N/mA = 4 cmt = 0.25 s
v = m/s
2.Springs & pendulumsHard
A 3.4 kg mass on a spring with k = 220 N/m is pulled 20 cm from equilibrium and released. Find its maximum acceleration amax.
m = 3.4 kgk = 220 N/mA = 20 cm
amax = m/s²
3.Waves, sound & opticsMedium
A pipe open at both ends is 1.2 m long. The speed of sound is 343 m/s. Find the frequency f2 of its harmonic number 2.
v = 343 m/sn = 2L = 1.2 m
f2 = Hz
4.Waves, sound & opticsHard
A sensor at rest sends ultrasound of 30 kHz towards a car that approaches at 20 m/s and mixes the echo with the emitted signal. Find the beat frequency fb (speed of sound 343 m/s).
f = 30 kHzu = 20 m/sv = 343 m/s
fb = Hz
3Solutions
1.Springs & pendulums
Answer: v = −0.2826 m/s
Simple harmonic motion: a mass m on a spring of stiffness k oscillates with angular frequency ω = √(k/m) and period T = 2π/ω = 2π·√(m/k). A simple pendulum of length L has T = 2π·√(L/g). With amplitude A, the largest speed is A·ω, the largest acceleration A·ω² and the energy ½·k·A².
Angular frequency
ω = √(190/2.4) = 8.898 rad/s
ω·t = 8.898·0.25 = 2.224 rad
Velocity
v = −A·ω·sin(ω·t) = −0.04·8.898·sin(2.224) = −0.2826 m/s
Answer
v = −0.2826 m/s
2.Springs & pendulums
Answer: amax = 12.94 m/s²
Simple harmonic motion: a mass m on a spring of stiffness k oscillates with angular frequency ω = √(k/m) and period T = 2π/ω = 2π·√(m/k). A simple pendulum of length L has T = 2π·√(L/g). With amplitude A, the largest speed is A·ω, the largest acceleration A·ω² and the energy ½·k·A².
Angular frequency
ω = √(220/3.4) = 8.044 rad/s
Maximum acceleration
amax = A·ω² = 0.2·8.044² = 12.94 m/s²
Answer
amax = 12.94 m/s²
3.Waves, sound & optics
Answer: f2 = 285.8 Hz
Standing waves and the Doppler effect: a string fixed at both ends, or a pipe open at both ends, fits whole half-wavelengths: L = nλ/2, fn = n·v/(2L), n = 1, 2, 3, … A pipe closed at one end has a node at the closed end and an antinode at the open end: L = nλ/4 with odd n only, fn = n·v/(4L). When source or observer move along the line between them, f′ = f·(v ± vo)/(v ∓ vs): use the upper signs for motion towards each other. Two waves of nearly equal frequency beat at |f₁ − f₂|.
Wavelength of this mode
λ = 2L/n = 2·1.2/2 = 1.2 m
Frequency
f = v/λ = 343/1.2 = 285.8 Hz
Answer
f2 = 285.8 Hz
4.Waves, sound & optics
Answer: fb = 3715 Hz
Standing waves and the Doppler effect: a string fixed at both ends, or a pipe open at both ends, fits whole half-wavelengths: L = nλ/2, fn = n·v/(2L), n = 1, 2, 3, … A pipe closed at one end has a node at the closed end and an antinode at the open end: L = nλ/4 with odd n only, fn = n·v/(4L). When source or observer move along the line between them, f′ = f·(v ± vo)/(v ∓ vs): use the upper signs for motion towards each other. Two waves of nearly equal frequency beat at |f₁ − f₂|.
Frequency at the car
f′ = 30000·(343 + 20)/343 = 31749 Hz
Frequency of the echo
f″ = f′·343/(343 − 20) = 33715 Hz
Beat frequency
fb = |f″ − f| = 3715 Hz
Answer
fb = 3715 Hz
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