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Resonance

Explore how periodic forces amplify vibrations. Sweep the driving frequency of a mechanical spring-mass oscillator to map its resonance response curve, observe energy transfer in Barton\'s coupled pendulums, and visualize pressure standing waves in acoustic tubes.

Resonance Laboratory Room ⓘ

Enable the simulation, adjust parameters, and observe peak response amplitude.

Driven Oscillation

Live Telemetry

Natural Freq (f0)
0.00 Hz
Driving Freq (fd)
0.00 Hz
Oscillation Amp
0.00 cm
Phase Difference
0.0°
Damping Ratio
0.00

Understanding Resonance in Physics

Resonance occurs when an oscillating system is subjected to an external driving force whose frequency matches or is close to one of the system\'s natural frequencies of vibration. When this matching condition occurs, the driving force does positive work on the system continuously, resulting in highly efficient energy transfer.

Consequently, the system\'s vibration amplitude increases dramatically, constrained only by the amount of damping (friction/drag) present.

The Resonance Condition

The natural angular frequency of a basic spring-mass oscillator is:

ω_0 = √(k / m)

When driven by a periodic external force F(t) = F_0 cos(ω t), the system reaches a steady-state oscillation where the amplitude A depends on the driving frequency ω:

A(ω) = (F_0 / m) / √[ (ω_0² - ω²)² + (2γω)² ]

Where γ = b / (2m) is the damping factor. If damping is zero (γ = 0) and the driving frequency matches the natural frequency (ω = ω_0), the amplitude theoretically goes to infinity. In the real world, damping limits this peak.

Damping and the Resonance Response Curve

Adjusting the damping coefficient (b) in the simulator alters the shape of the resonance curve:

  • Low Damping: Creates a very tall, narrow peak. The system responds strongly, but only to a narrow band of frequencies.
  • High Damping: Flattens and widens the resonance peak. The maximum amplitude is much smaller, and the peak frequency shifts slightly to the left (lower frequency).
  • Phase Shift (φ): Below resonance, the oscillator moves in-phase with the driving force (φ ≈ 0). At resonance, the phase difference is exactly 90° (π/2 radians), meaning the force pushes at the moment of maximum velocity. Well above resonance, the oscillator moves completely out-of-phase (φ ≈ 180°).

Barton\'s Coupled Pendulums

Barton\'s Pendulums represent a classic demonstration of coupled resonance. A heavy driver pendulum is suspended from a support string alongside several bobs of different lengths:

  • Since length dictates natural period (T = 2π√(L/g)), bobs of different lengths have different natural frequencies.
  • When the heavy driver pendulum is released, its swing periodically pulls on the support string, acting as a driving force.
  • The bob that matches the driver pendulum\'s length swings with the largest amplitude, demonstrating resonance. Bobs that are much longer or shorter oscillate with negligible amplitude because they are driven off-resonance.

Acoustic Resonance in Air Columns

Resonance also governs sound waves inside pipes. An air column closed at one end (like a water cylinder) forms standing waves when excited by a sound source. Since the closed end is a displacement node (air cannot move) and the open mouth is a displacement antinode (air moves freely), resonance occurs only when the column length L fits an odd number of quarter wavelengths:

L = n · λ / 4 \quad (n = 1, 3, 5, \dots)

Using the relationship between wave speed, frequency, and wavelength (v = f · λ), the resonant frequencies are:

f_n = n · v / (4L)

In our third simulator mode, you can adjust the water level to change the air column length L, tuning it to fundamental (n=1) or harmonic resonances (n=3, n=5) for a given tuning fork pitch.

Solved Examples

Example 1

A driven spring-mass system consists of a mass m = 0.5 kg and a spring constant k = 200 N/m. The driving force is described by F(t) = F_0 cos(ωt). (a) Calculate the natural angular frequency ω_0 and natural frequency f_0 of the system. (b) If the driving frequency is tuned to exactly 3.18 Hz, what behavior do you expect, and why?

View Step-by-Step Solution
  1. Part (a): Identify the parameters: mass m = 0.5 kg, spring constant k = 200 N/m.
  2. Use the natural angular frequency formula: ω_0 = √(k / m).
  3. ω_0 = √(200 / 0.5) = √(400) = 20 rad/s.
  4. Calculate natural frequency: f_0 = ω_0 / 2π.
  5. f_0 = 20 / (2 · 3.14159) ≈ 3.183 Hz.
  6. Part (b): Note that the driving frequency f_d = 3.18 Hz is extremely close to the natural frequency f_0 = 3.183 Hz.
  7. Because the driving frequency matches the natural frequency (f_d ≈ f_0), the system undergoes **Resonance**.
  8. Assuming low damping, the oscillation amplitude will grow very large, as energy from the driver is transferred in-phase with the mass's velocity.
  9. The natural frequency is 3.18 Hz. At a driving frequency of 3.18 Hz, the system undergoes resonance, causing large-amplitude oscillations.

**Final Answer:** f_0 = 3.18 Hz, system enters Resonance

Example 2

An acoustic resonance tube closed at one end is excited by a tuning fork vibrating at f = 512 Hz. Taking the speed of sound in air to be v = 343 m/s, (a) calculate the wavelength of the sound wave. (b) Find the shortest length L of the air column that will produce resonance (fundamental mode).

View Step-by-Step Solution
  1. Part (a): Use the wave speed equation: v = f · λ.
  2. Solve for wavelength λ: λ = v / f.
  3. λ = 343 / 512 ≈ 0.670 meters (or 67.0 cm).
  4. Part (b): For an air column closed at one end, the resonance condition is L = nλ / 4 (for odd integers n = 1, 3, 5, ...).
  5. The shortest resonant length corresponds to the fundamental mode (n = 1).
  6. L = λ / 4 = 0.670 / 4 = 0.1675 meters (or 16.75 cm).
  7. The wavelength is 67.0 cm and the shortest resonant column length is 16.75 cm.

**Final Answer:** λ = 67.0 cm, L = 16.8 cm

Example 3

A driven, damped harmonic oscillator is subjected to an external driving force. The natural frequency is ω_0 = 10 rad/s, the damping coefficient is γ = 1 rad/s, and the driving force amplitude per unit mass is F_0/m = 10 N/kg. Calculate the steady-state amplitude of oscillation when (a) the driving frequency is ω = 2 rad/s (well below resonance), (b) ω = 10 rad/s (at natural frequency), and (c) ω = 20 rad/s (well above resonance).

View Step-by-Step Solution
  1. Use the steady-state amplitude formula: A(ω) = (F_0/m) / √[ (ω_0² - ω²)² + (2γω)² ].
  2. Part (a): For ω = 2 rad/s:
  3. Denominator term 1: (ω_0² - ω²)² = (10² - 2²)² = (100 - 4)² = 96² = 9216.
  4. Denominator term 2: (2γω)² = (2 · 1 · 2)² = 4² = 16.
  5. A(2) = 10 / √(9216 + 16) = 10 / √(9232) ≈ 10 / 96.08 ≈ 0.104 meters (10.4 cm).
  6. Part (b): For ω = 10 rad/s (resonance):
  7. Denominator term 1: (ω_0² - ω²)² = (100 - 100)² = 0.
  8. Denominator term 2: (2γω)² = (2 · 1 · 10)² = 20² = 400.
  9. A(10) = 10 / √(0 + 400) = 10 / 20 = 0.500 meters (50 cm).
  10. Part (c): For ω = 20 rad/s:
  11. Denominator term 1: (ω_0² - ω²)² = (100 - 400)² = (-300)² = 90000.
  12. Denominator term 2: (2γω)² = (2 · 1 · 20)² = 40² = 1600.
  13. A(20) = 10 / √(90000 + 1600) = 10 / √(91600) ≈ 10 / 302.65 ≈ 0.033 meters (3.3 cm).
  14. Comparing the amplitudes: A(2) = 10.4 cm, A(10) = 50.0 cm (maximum at resonance), A(20) = 3.3 cm.
  15. The amplitudes are 10.4 cm (low frequency), 50.0 cm (resonance), and 3.3 cm (high frequency).

**Final Answer:** A(2) = 10.4 cm, A(10) = 50 cm, A(20) = 3.3 cm

Common Misconceptions & Pitfalls

  • Misconception: Resonance happens at any frequency as long as the force is strong.
    **Reality:** No. If the force is applied at a frequency far from the natural frequency, the work done on the oscillator alternates between positive and negative, resulting in negligible net energy transfer. The driving force must match the natural frequency.
  • Misconception: Resonant frequency is exactly the same as natural frequency under all conditions.
    **Reality:** Only in undamped systems. In damped systems, the frequency at which the maximum amplitude occurs (the peak of the curve) is slightly lower than the natural frequency, given by ω_res = √(ω_0² - 2γ²).
  • Misconception: Open and closed pipes resonate at the exact same lengths for a given pitch.
    **Reality:** A pipe open at both ends requires antinodes at both ends, making the fundamental resonant length L = λ/2. A closed pipe requires a node at the closed end, making its fundamental length L = λ/4 (half the length of the open pipe for the same note).

Practice Questions

Question 1

What is the phase relationship between the driving force and the displacement of the oscillator at resonance, and why does this lead to maximum energy transfer?

Show Explanation

At resonance, the phase difference between the driving force and the displacement is exactly 90° (π/2 radians) or, equivalently, the driving force is perfectly in-phase with the velocity of the oscillator. Power is defined as force times velocity (P = F · v). Because the force pushes in the exact same direction the mass is already moving at all points in the cycle, the work done on the system is positive throughout the entire cycle. This results in maximum rate of energy transfer, causing the amplitude to grow to its limit.

Question 2

In Barton's Pendulum experiment, why does the bob with the same length as the driver swing with the largest amplitude, while the others remain almost still?

Show Explanation

The time period of a simple pendulum is T = 2π√(L/g), meaning its natural frequency depends solely on its length. The driver pendulum swings back and forth at its own natural frequency, periodically shaking the support string at that specific frequency. For bobs suspended from the same string: (1) The bob with length equal to the driver has the same natural frequency, matching the driving rate. It experiences resonance, absorbing energy efficiently. (2) Bobs with different lengths have different natural frequencies. They are forced to oscillate away from their natural frequency, resulting in small amplitudes and phase mismatches.

Question 3

Why do soldiers break step when crossing a bridge, and how does this relate to mechanical resonance?

Show Explanation

When soldiers march in step, they strike the ground with a periodic force at a specific frequency. Every bridge has natural frequencies of vibration. If the marching frequency matches one of the bridge's natural frequencies, resonance occurs. The energy of the soldiers' steps is continuously transferred into the bridge's structure, creating expanding standing wave vibrations. This can lead to structural damage or complete collapse (as occurred to the Broughton Suspension Bridge in 1831). Breaking step randomizes the impact frequencies, preventing constructive resonance.

Question 4

How does magnetic resonance imaging (MRI) utilize the concept of resonance in medical diagnostics?

Show Explanation

MRI uses strong magnetic fields to align the spin vectors of hydrogen nuclei (protons) in water molecules inside body tissues. A radiofrequency (RF) pulse is then applied at a specific frequency (the Larmor frequency), which matches the natural precession frequency of the aligned protons. This causes the protons to absorb the RF energy (resonance) and flip their spin directions. When the RF pulse is turned off, the protons relax back to their aligned state, emitting the absorbed energy as radio signals. The timing and strength of these return signals are mapped to generate high-resolution cross-sectional images of the tissues.

Frequently Asked Questions

What is resonance?
Resonance is a physical phenomenon where an oscillating system is driven by an external periodic force at a frequency equal or close to the system's natural frequency, resulting in large amplitude vibrations.
What is the difference between free oscillation and forced oscillation?
Free oscillation occurs when a system is displaced and allowed to vibrate on its own at its natural frequency. Forced oscillation occurs when a system is continuously driven by an external periodic force, forcing it to vibrate at the frequency of the driver.
What is a resonance curve?
A resonance curve (or response curve) is a plot of the steady-state amplitude of a forced oscillator on the vertical axis against the driving frequency on the horizontal axis. It peaks at the natural frequency.
How does damping affect the resonance curve?
Higher damping reduces the peak amplitude at resonance, widens the curve, and shifts the peak slightly to a lower frequency. Lower damping creates a taller, sharper peak.
What is Barton's Pendulum?
Barton's Pendulum is a demonstration of mechanical resonance where a heavy driving pendulum excites several bobs of different lengths hung from a common string. Only the bob matching the driver's length resonates.
What is acoustic resonance?
Acoustic resonance occurs when an air column (like inside a pipe or bottle) vibrates at a natural frequency matching an exciting sound source (like a tuning fork), amplifying the sound through standing waves.