Energy, damping and resonance flashcards

Test yourself one card at a time: say the answer, flip, then mark it. Cards you know drop out; the rest come back until you know them. Your progress stays in this browser.

Free oscillation
An oscillation with no external periodic force acting; the system oscillates at its natural frequency.
Forced oscillation
An oscillation driven by an external periodic force; in the steady state the system oscillates at the driving frequency.
Natural frequency f₀
The frequency at which a system oscillates when displaced and then left to oscillate freely.
Resonance
The large increase in amplitude of a forced oscillation when the driving frequency equals the natural frequency; energy is transferred most efficiently.
Damping
A force opposing the motion of an oscillator that transfers energy out of the system (usually to thermal energy), so the amplitude decreases.
Critical damping
The amount of damping that returns a displaced system to equilibrium in the shortest time without oscillating.
Light and heavy damping
Light: oscillates with slowly decreasing amplitude. Heavy: no oscillation, but returns to equilibrium more slowly than critical damping.
Ductile material (in damping)
A material that can undergo large plastic deformation, repeatedly, without fracturing — so it can dissipate energy each cycle.
Period of a mass–spring oscillator
T = 2π√(m/k); f = 1/T. (Yes)
Total energy of a mass–spring oscillator
E = ½kA² (strain energy at the extreme). (No — derive it)
Kinetic energy at displacement x
E_k = ½k(A² − x²) = total − ½kx². (No — derive it)
Energy and amplitude, any oscillator
E ∝ A²: doubling A gives 4E; halving E gives A/√2. (No — learn it)
Core Practical 16 straight-line form
T² = (4π²/k)m, so gradient of T² against m = 4π²/k. (No — derive it)
Energy dissipated per cycle by plastic deformation
Area enclosed by the force–extension loop. (No — learn it)
Energy against displacement for an undamped oscillator
PE: parabola, zero at x = 0, maximum at ±A. KE: inverted parabola, maximum at x = 0. Total: horizontal line.
Energy against time for an undamped oscillator
KE and PE both vary at twice the oscillation frequency, never negative, out of step; their sum is constant.
Displacement–time graph with light damping
Oscillation at almost constant period inside an exponentially decaying envelope; amplitude falls by the same fraction each cycle.
Resonance curve (amplitude against driving frequency)
Small at low frequency, sharp peak at (or just below) f₀, very small at high frequency.
Effect of more damping on the resonance curve
Peak lower, broader and at a slightly lower frequency; little change far from f₀.
Core Practical 16 apparatus
Spring hung from a vibration generator driven by a signal generator; slotted masses; metre rule and pointer to judge amplitude.
Why does a forced oscillation reach a steady amplitude?
The amplitude grows until the energy supplied by the driver each cycle equals the energy removed by damping each cycle.
Why is the energy transfer greatest at resonance?
The driving force is in step with the motion, so it does positive work on the oscillator throughout each cycle.
How do car shock absorbers work?
Oil is forced through small holes; viscous drag transfers the oscillation energy to thermal energy. Damping is near critical.
Three ways a building in an earthquake zone limits its oscillation
Natural frequency kept away from earthquake frequencies (e.g. base isolators); dampers remove energy; ductile steel deforms plastically, dissipating energy.
Why must earthquake dampers be ductile, not brittle?
Ductile metal can deform plastically many times, dissipating energy each cycle; a brittle material would crack and absorb little energy.
Core Practical 16: why is there an intercept on the T² axis?
Part of the spring's own mass oscillates with the load, adding an effective extra mass.
Core Practical 16: how is the resonant frequency found?
Increase the signal-generator frequency in small steps and find where the amplitude of the mass is greatest; repeat from above and average.
m = 0.40 kg, k = 25 N m⁻¹, A = 6.0 cm. Total energy and maximum speed?
E = ½ × 25 × 0.060² = 0.045 J; v_max = √(2 × 0.045 ÷ 0.40) = 0.47 m s⁻¹.
Damping reduces the amplitude to 80% of its starting value. Fraction of energy left?
0.80² = 0.64, so 64% of the energy remains (36% lost).
Car body 1400 kg on suspension k = 2.0 × 10⁵ N m⁻¹; humps 6.0 m apart. Resonant speed?
f₀ = (1/2π)√(k/m) = 1.9 Hz; v = f₀ × 6.0 = 11 m s⁻¹.
T² against m has gradient 1.97 s² kg⁻¹. Spring constant?
k = 4π² ÷ 1.97 = 20 N m⁻¹.
Oscillation energy 1.97 × 10⁴ J at amplitude 0.10 m; dampers remove 3.2 kJ. New amplitude?
A = 0.10 × √(16.5 ÷ 19.7) = 0.092 m.
Amplitude and energy ratios
Square an amplitude ratio to get the energy ratio; square-root an energy ratio to get the amplitude ratio.
Defining resonance
Name both: driving frequency = natural frequency; then maximum amplitude and maximum energy transfer. “Frequencies match” alone is too vague.
Critical vs heavy damping
Both avoid oscillation; critical is the quickest return to equilibrium. Always state both parts.
Using √(k/m) as a frequency
√(k/m) is ω in rad s⁻¹; divide by 2π for f in Hz.
Where does damped energy go?
Say it: transferred from kinetic/potential energy of the oscillator to thermal energy of the surroundings or damper.

Practise this topicNotes Questions (40)Test builder