Vibrations and Oscillations

23.1 What Makes Motion Oscillatory

Most motion in nature does not repeat. A thrown ball follows a parabolic arc and lands. A car accelerates, cruises, and stops. These are one-way, non-repeating motions. Oscillatory motion is fundamentally different: the system moves, reverses, returns, and then repeats the same sequence indefinitely, or until energy is lost.

Three conditions produce oscillatory motion. First, the system must have an equilibrium position — a configuration toward which it naturally tends. Second, when displaced from equilibrium, the system must experience a restoring force directed back toward that equilibrium. Third, the system must have inertia: it carries momentum through the equilibrium position and overshoots to the other side, where the restoring force again pushes it back. It is the interplay of restoring force and inertia that sustains repetition.

Oscillation is not merely back-and-forth motion. It is a specific physical process in which energy cycles continuously between two forms: kinetic energy, associated with velocity through the equilibrium position, and potential energy, associated with displacement against the restoring force. Understanding this energy exchange is the key to understanding every oscillating system, from a pendulum to a quartz oscillator to the vibrating air column in a flute.

23.2 Periodic Motion and Repetition

Periodic motion is any motion that repeats itself at regular time intervals. Oscillatory motion is always periodic. Not all periodic motion is oscillatory in the strict sense — a ball rolling in a circular track repeats, but it does not oscillate around an equilibrium — but for the purposes of this chapter, we focus on the oscillatory case.

The key feature of periodic motion is the period \(T\): the time required to complete one full cycle. After exactly one period, the system is back in the same position moving in the same direction with the same velocity as it started. Every measurable property of the system — position, velocity, acceleration, potential energy, kinetic energy — repeats with the same period \(T\).

Periodicity is an idealization. Real oscillating systems lose energy to friction, air resistance, or internal damping, so the amplitude gradually decreases. But the period remains nearly constant even as energy is lost — a critically important property that makes oscillating systems useful as timekeepers. A pendulum clock keeps time not because the pendulum swings far, but because it swings regularly.

23.3 Equilibrium Position in Oscillating Systems

The equilibrium position is the configuration of the system where the net force is zero. It is the position the system naturally adopts when undisturbed. Crucially, the equilibrium position for an oscillating system is a stable minimum in potential energy: the system naturally returns to it if displaced by a small amount, but it will not necessarily stay there because it arrives with kinetic energy gained during the return journey.

For a mass hanging on a spring, the equilibrium position is where the spring force exactly balances gravity. For a simple pendulum, it is the lowest point directly below the pivot. For a floating object, it is the level at which the buoyancy force equals the object’s weight. In each case, displacing the system from equilibrium creates a net force or torque directed back toward equilibrium.

A common misconception is that the system “wants to stay” at the equilibrium position. In fact, an undamped system passes through equilibrium at maximum speed and maximum kinetic energy — it cannot stop there unless energy is removed. The equilibrium is the point of minimum potential energy, but the total mechanical energy is conserved, so kinetic energy is at its maximum precisely when potential energy is at its minimum.

23.4 Amplitude, Period, and Frequency

Four quantities fully describe the kinematics of a periodic oscillation: amplitude, period, frequency, and phase. At this level we focus on the first three.

Amplitude \(A\) is the maximum displacement from the equilibrium position. It measures how far the system swings or stretches in each direction. Amplitude is always a positive quantity. The system reaches \(+A\) on one side and \(-A\) on the other. Amplitude is determined entirely by initial conditions — how hard the system is displaced or pushed at the start.

Period \(T\) is the time for one complete cycle, measured in seconds. One complete cycle is one full excursion from the starting point, across to the far side, and back to the starting point moving in the original direction.

Frequency \(f\) is the number of complete cycles per second, measured in hertz (Hz), where \(1\text{ Hz}=1\) cycle per second. A tuning fork vibrating at 440 Hz completes 440 full oscillations every second. The human ear detects frequencies roughly from 20 Hz to 20,000 Hz. Ultrasound used in medical imaging operates at millions of hertz.

23.5 Period–Frequency Relationship

Period and frequency are reciprocals of each other. If a system completes \(f\) cycles per second, each cycle takes \(1/f\) seconds. Conversely, if each cycle takes \(T\) seconds, the system completes \(1/T\) cycles per second.

\[ f = \frac{1}{T} \qquad T = \frac{1}{f} \]

This reciprocal relationship is universal — it applies to every oscillating system without exception. A pendulum with a 2-second period has a frequency of 0.5 Hz. A guitar A-string vibrating at 440 Hz has a period of \(1/440 \approx 0.00227\text{ s}=2.27\text{ ms}\). A power-grid oscillation at 60 Hz has a period of \(1/60 \approx 0.0167\text{ s}\).

Angular frequency \(\omega\) is an alternative way to express frequency used in mathematical treatments of oscillation. It equals \(2\pi f\), or equivalently \(2\pi/T\), and has units of radians per second.

\[ \omega = 2\pi f = \frac{2\pi}{T} \]

23.6 Simple Harmonic Motion

Simple harmonic motion (SHM) is a specific, mathematically defined form of oscillatory motion in which the restoring force is directly proportional to displacement from equilibrium and always directed toward equilibrium. SHM is the simplest and most important type of oscillation in physics.

In SHM, the position of the oscillating object as a function of time is a sinusoidal function — a sine or cosine curve. The object moves smoothly between its maximum positive and negative displacements, passing through equilibrium at maximum speed and slowing to momentary rest at the turning points.

\[ x(t) = A\cos(\omega t+\phi) \]

The velocity in SHM is the derivative of position with respect to time, giving \(v(t)=-A\omega\sin(\omega t+\phi)\). The maximum speed at equilibrium is \(v_{\max}=A\omega=2\pi fA\). The acceleration is the derivative of velocity: \(a(t)=-A\omega^2\cos(\omega t+\phi)=-\omega^2x(t)\). This last result is fundamental: in SHM, acceleration is always proportional to displacement and oppositely directed.

\[ a = -\omega^2 x \]

The negative sign is crucial. When displacement is positive, acceleration is negative; when displacement is negative, acceleration is positive. This self-correcting behavior is what sustains the oscillation indefinitely in the absence of damping.

23.7 Restoring Forces

A restoring force is any force that acts to return a displaced system toward its equilibrium position. The defining property of a restoring force is that it is directed toward equilibrium — it always opposes the displacement.

The simplest restoring force is proportional to displacement: \(F=-kx\), where \(k\) is the stiffness constant of the system and \(x\) is the displacement from equilibrium. The negative sign signals that the force is always opposite to the displacement. This linear restoring force is Hooke’s Law, and systems obeying it produce perfect SHM.

Many restoring forces are approximately linear for small displacements even if they are nonlinear for large ones. A pendulum’s restoring force is \(F=mg\sin\theta\), which is nonlinear. But for small angles, \(\sin\theta \approx \theta\) when \(\theta\) is measured in radians, making the restoring force approximately linear and the motion approximately SHM.

Restoring forces arise in many physical systems. In a spring, it is the elastic force. In a pendulum, it is the component of gravity tangent to the arc. In a floating object, it is the excess or deficit of buoyancy when displaced vertically. In a molecular bond, it is the interatomic electromagnetic force.

23.8 Mass–Spring Oscillators

The mass–spring system is the prototype oscillator of classical mechanics. It consists of a mass \(m\) attached to a spring with spring constant \(k\). When displaced from equilibrium by a distance \(x\), the spring exerts a restoring force \(F=-kx\). By Newton’s second law, \(F=ma\), giving \(ma=-kx\), or \(a=-(k/m)x\). Comparing this to the SHM condition \(a=-\omega^2x\), we identify:

\[ \omega = \sqrt{\frac{k}{m}} \qquad T = 2\pi\sqrt{\frac{m}{k}} \qquad f = \frac{1}{2\pi}\sqrt{\frac{k}{m}} \]

This result has three remarkable features. First, the period is independent of amplitude as long as the spring obeys Hooke’s Law. Second, increasing the mass increases the period. Third, increasing the spring constant decreases the period. A heavier mass has more inertia and responds more slowly; a stiffer spring has a stronger restoring force and pulls the mass back faster.

The mass–spring system can be oriented in any direction. For a vertical spring, gravity shifts the equilibrium position downward but does not change the period, because the oscillation occurs around the new equilibrium and gravity cancels in the equation of motion for displacement from equilibrium.

23.9 Pendulum Motion

A simple pendulum consists of a small mass, called the bob, on a massless, inextensible string of length \(L\), free to swing about a fixed pivot. When displaced to a small angle \(\theta\) from vertical, the component of gravity tangent to the arc provides the restoring force: \(F_{\text{tangent}} = mg\sin\theta \approx mg\theta\) for small angles.

\[ T = 2\pi\sqrt{\frac{L}{g}} \qquad f = \frac{1}{2\pi}\sqrt{\frac{g}{L}} \]

Three things are notable about this result. First, the period is independent of the mass of the bob. Second, the period depends on \(g\), so a pendulum on the Moon would swing more slowly than the same pendulum on Earth. Third, the period increases with length, so a longer pendulum swings more slowly.

Pendulum motion deviates from the simple formula for large amplitudes. When the angle exceeds about 15°, the small-angle approximation fails and the period begins to increase noticeably. At 30° amplitude, the true period is about 1.7% longer than the simple formula predicts.

23.10 Energy in Oscillating Systems

One of the most powerful ways to analyze oscillatory motion is through energy. In an ideal undamped oscillator, mechanical energy is conserved: it does not increase or decrease, but is continuously converted between kinetic and potential form.

For a mass–spring oscillator, the potential energy stored in the spring when it is displaced by \(x\) is \(U=\frac{1}{2}kx^2\). The kinetic energy of the mass moving at speed \(v\) is \(K=\frac{1}{2}mv^2\). At maximum displacement \(A\), velocity is zero, so all energy is potential. At equilibrium, all energy is kinetic.

\[ E = \frac{1}{2}kA^2 = \frac{1}{2}kx^2 + \frac{1}{2}mv^2 = \text{constant} \]

This energy equation lets us find the velocity at any position without needing the time-dependent equations. The maximum speed is \(v_{\max}=A\sqrt{k/m}=A\omega\). The total energy is proportional to the square of the amplitude: doubling the amplitude quadruples the energy.

23.11 Damping and Energy Loss

All real oscillators lose energy over time. The mechanism may be friction between surfaces, air resistance, viscous drag in a fluid, or internal molecular rearrangement within the material. Any process that dissipates mechanical energy into heat is a form of damping.

Damping has two effects. First, it reduces the amplitude of oscillation over time. Second, it slightly changes the period, though for lightly damped systems this effect is negligible. A lightly damped oscillator maintains its periodic character for many cycles before the amplitude becomes insignificant. A heavily damped oscillator returns to equilibrium without completing even one full oscillation.

Damping is not always unwanted. Car shock absorbers are deliberately designed to be critically damped: they allow the suspension to absorb a bump and return to normal position as quickly as possible without bouncing. A door closer, a galvanometer needle, and a bathroom scale all use deliberate damping to produce smooth, non-oscillatory behavior.

23.12 Driven Oscillations

A driven oscillator is one subjected to a periodic external force in addition to its restoring force. The driving force supplies energy to the system, compensating for damping and maintaining oscillation. If the driving force is absent, the oscillator gradually decays due to damping. With driving, the system can sustain indefinite oscillation.

The behavior of a driven oscillator depends critically on the relationship between the driving frequency \(f_d\) and the natural frequency \(f_0\), the frequency the system would oscillate at without driving. When these frequencies differ significantly, the system oscillates at the driving frequency with modest amplitude. When the driving frequency approaches the natural frequency, the situation changes dramatically.

The natural frequency, or resonant frequency, of a system is determined entirely by its physical properties: the mass and spring constant for a mass–spring system, or the length and gravity for a pendulum. It is the frequency at which the system “wants” to oscillate. When a driver applies energy at this frequency, the system absorbs energy most efficiently, and the amplitude builds to a maximum.

The phase relationship between the driver and the oscillator also depends on the driving frequency. Below the natural frequency, the oscillator is nearly in phase with the driver. At the natural frequency, the oscillator lags the driver by exactly 90°. Above the natural frequency, the oscillator is nearly 180° out of phase.

23.13 Resonance (Introductory)

Resonance is the phenomenon in which a system oscillates with maximum amplitude when driven at its natural frequency. The driving force and the natural oscillation reinforce each other cycle after cycle, and amplitude grows until it is limited only by damping. In a lightly damped system, resonance can produce amplitudes vastly larger than those the driving force alone could cause.

Resonance explains how a singer’s voice can shatter a crystal glass, how radio receivers select a single station from thousands, how musical instruments amplify specific frequencies, and how bridges can fail catastrophically when wind or traffic drives them near their natural frequencies.

Adapted from engineering lore

The collapse of the Tacoma Narrows Bridge in 1940 is one of the most dramatic examples of resonance disaster. Wind-induced oscillations drove the bridge at a frequency near its torsional natural frequency, producing growing oscillations until the bridge twisted itself apart. Modern bridge design explicitly accounts for aeroelastic resonance by designing structures whose natural frequencies are far removed from any expected driving frequencies.

Beneficial resonance is equally important. MRI machines use nuclear magnetic resonance. Laser cavities resonate at optical frequencies, producing coherent light. Musical resonance makes instruments expressive and responsive.

23.14 Common Mistakes in Oscillation Analysis

Students learning about oscillatory motion make predictable errors. Awareness of these mistakes helps avoid them.

• Confusing period and frequency. Period is in seconds; frequency is in hertz. They are reciprocals.
• Assuming amplitude affects the period of a spring or pendulum. For an ideal spring and for a small-angle pendulum, the period is independent of amplitude.
• Forgetting that the pendulum formula requires radians and small angles. The formula \(T=2\pi\sqrt{L/g}\) is valid only for small amplitudes.
• Confusing the equilibrium position with the position of zero velocity. The equilibrium position is where velocity is maximum. The positions of zero velocity are the turning points.
• Forgetting that energy scales as \(A^2\), not \(A\). Doubling the amplitude quadruples the energy.
• Believing that damping significantly changes the period. For light to moderate damping, the period is nearly unchanged.
• Confusing natural frequency with driving frequency. Resonance occurs when they match.
• Treating resonance as always dangerous. Resonance can be beneficial or harmful depending on context.

Chapter Summary

Oscillatory motion is repetitive motion around an equilibrium position, sustained by the interplay of a restoring force and inertia. It underlies all wave phenomena: waves occur when oscillation propagates through space.

• For oscillation to occur: an equilibrium position must exist, displacement must create a restoring force toward equilibrium, and the system must have inertia that carries it through equilibrium.
• Period \(T\) is the time for one complete cycle; frequency \(f\) is the number of cycles per second; they are reciprocals: \(f=1/T\).
• Angular frequency \(\omega=2\pi f=2\pi/T\) measures the rate of phase advance in radians per second.
• Simple harmonic motion occurs when the restoring force is proportional to displacement. Position varies sinusoidally: \(x(t)=A\cos(\omega t+\phi)\).
• Mass–spring: \(\omega=\sqrt{k/m}\); \(T=2\pi\sqrt{m/k}\). Period depends on \(m\) and \(k\), not amplitude.
• Simple pendulum, small angle: \(T=2\pi\sqrt{L/g}\). Period depends on \(L\) and \(g\), not mass or amplitude.
• Total energy of an oscillator equals \(\frac{1}{2}kA^2\); it is proportional to \(A^2\) and conserved in the absence of damping.
• Damping dissipates energy, reducing amplitude over time. Critically damped systems return to equilibrium fastest without oscillating.
• Driven oscillators are sustained by periodic external forces. Resonance occurs when the driving frequency matches the natural frequency, producing maximum amplitude.

Key Terms

oscillatory motion

Repetitive motion in which a system moves back and forth around an equilibrium position under the action of a restoring force.

periodic motion

Any motion that repeats itself at regular time intervals; characterized by a well-defined period \(T\).

equilibrium position

The configuration of an oscillating system at which the net force is zero; the position around which oscillation occurs.

amplitude (\(A\))

The maximum displacement from equilibrium; determines the extent of the oscillation and is set by initial conditions.

period (\(T\))

The time required for one complete oscillation cycle; measured in seconds.

frequency (\(f\))

The number of complete oscillation cycles per second; measured in hertz; \(f=1/T\).

angular frequency (\(\omega\))

The rate of phase change of the oscillation; \(\omega=2\pi f\); measured in radians per second.

simple harmonic motion (SHM)

Oscillatory motion in which the restoring force is proportional to displacement; produces sinusoidal position vs. time; described by \(a=-\omega^2x\).

restoring force

A force directed toward the equilibrium position, whose magnitude is proportional to displacement for SHM: \(F=-kx\).

spring constant (\(k\))

The stiffness of a spring; the proportionality constant in Hooke’s Law; measured in N/m.

natural frequency (\(f_0\))

The frequency at which a system oscillates freely when displaced; determined by the physical properties of the system.

damping

Any process that dissipates mechanical energy from an oscillating system, reducing amplitude over time.

driven oscillator

An oscillating system subjected to a periodic external driving force that compensates for energy loss and sustains oscillation.

resonance

The condition in which a system is driven at its natural frequency, producing maximum amplitude of oscillation.

isochronism

The property of an oscillator, such as a small-angle pendulum, in which the period is independent of amplitude.

Concept Review Questions

Answer in complete sentences. No calculations required.

1. What three conditions are necessary for a system to oscillate? Explain the role of each.
2. Explain why an oscillating system passes through its equilibrium position at maximum speed rather than coming to rest there.
3. A pendulum bob is pulled to one side and released. Describe the energy transformations that occur during one complete oscillation.
4. Why is the period of a mass–spring oscillator independent of amplitude? Use energy reasoning in your answer.
5. How does the period of a simple pendulum change if: (a) the mass is doubled, (b) the string length is quadrupled, (c) the pendulum is taken to the Moon?
6. What is the difference between natural frequency and driving frequency? Under what condition does resonance occur?
7. Explain what critical damping means and give a practical example of where it is deliberately engineered.
8. Why is resonance sometimes beneficial and sometimes catastrophically harmful? Give one example of each.

Oscillation Problems

Show all work. Include units throughout. Round appropriately.

9. An object oscillates with period \(T=0.400\text{ s}\). Find its frequency and angular frequency.
10. A tuning fork vibrates at 512 Hz. Find its period in milliseconds.
11. A mass–spring system has \(m=0.250\text{ kg}\) and \(k=36.0\text{ N/m}\). Find (a) the angular frequency, (b) the period, and (c) the frequency.
12. A spring oscillator has \(T=1.20\text{ s}\). If the mass is increased by a factor of 4, with spring constant unchanged, what is the new period?
13. A spring oscillator has \(f=2.00\text{ Hz}\) with a 0.500 kg mass. Find the spring constant \(k\).
14. An oscillator has amplitude \(A=0.0800\text{ m}\) and spring constant \(k=50.0\text{ N/m}\). Find (a) the total energy, (b) the maximum speed, and (c) the speed when \(x=0.0400\text{ m}\).
15. A mass on a spring oscillates with maximum speed 0.600 m/s and amplitude 0.0300 m. Find the angular frequency and spring constant if \(m=0.100\text{ kg}\).

Frequency and Period Exercises

16. The human heart at rest beats about 72 times per minute. Find the period in seconds and frequency in Hz of the heartbeat.
17. A metronome is set to 120 beats per minute. Find the period of each beat and the frequency in Hz.
18. Complete the table:

    Problem 18 Period, frequency, and angular frequency.

    Period \(T\) (s)	Frequency \(f\) (Hz)	Angular Frequency \(\omega\) (rad/s)
    0.500	?	?
    ?	440	?
    0.020	?	?
    ?	?	100

19. A clock pendulum completes 3600 full oscillations per hour. What is its period? Its frequency?
20. Radio station KPHP broadcasts at 98.7 MHz. Find the period of one oscillation of the broadcast carrier wave.

Spring and Pendulum Analysis

21. Find the length of a simple pendulum with a period of 1.00 s on Earth, where \(g=9.81\text{ m/s}^2\).
22. A pendulum has length 0.625 m. Find its period on (a) Earth, (b) the Moon, and (c) Mars.
23. A grandfather clock uses a 0.993 m pendulum on Earth. It is then used on a space station where \(g_{\text{eff}}=4.90\text{ m/s}^2\). By how many minutes per hour will the clock gain or lose?
24. A spring with \(k=80.0\text{ N/m}\) and mass 0.320 kg oscillates with amplitude 5.00 cm. Find (a) the period, (b) the maximum kinetic energy, and (c) the position at which \(KE=PE\).
25. Two springs each with \(k=40.0\text{ N/m}\) are connected to a 0.200 kg mass: (a) in series, where \(k_{\text{series}}=k/2\), and (b) in parallel, where \(k_{\text{parallel}}=2k\). Find the period in each case.
26. A mass–spring oscillator is constructed to have a period of exactly 0.500 s using a 0.400 kg mass. What spring constant is required?

Resonance Concept Problems

27. A crystal wine glass has a natural frequency of 400 Hz. A singer sustains a note at 390 Hz near the glass. Will the glass resonate? Explain what would happen if the singer gradually raised the pitch to 400 Hz.
28. A child pumps a swing at a frequency of 0.5 Hz, which is the natural frequency of the swing. Explain why the amplitude builds with each push, even though each individual push is small.
29. A building is 20 stories tall and has a natural oscillation period of about 4 seconds. Earthquake P-waves arrive at 2 Hz. Are these waves likely to drive the building into resonance? Explain.
30. A car with worn shock absorbers tends to bounce at about 1.2 Hz on a rough road. At what road spacing would the bouncing become most severe if the car is traveling at 60 km/h? Hint: find the road-feature period at that speed and spacing.
31. Explain why radio tuning selects one station from thousands. What physical quantity of the receiver’s circuit is adjusted, and to what value is it matched?

Chapter 23 Checkpoint

Before moving on to Chapter 24, confirm you can do each of the following without notes.