Oscillations

The One Thing

Simple harmonic motion (SHM) is oscillatory motion in which the restoring acceleration is directly proportional to displacement and directed toward the stable equilibrium position: $a=-{\omega }^{2}x$. Its timing, motion and energy follow from this restoring condition and, in ideal cases, conservation of mechanical energy.

Reactions, Processes and Experiments

Key Terms

• Periodic motion: Motion that repeats itself after equal intervals of time.
• Oscillatory motion: Repetitive motion in which an object moves to and fro about a mean or equilibrium position.
• Equilibrium position: The position where the net force or net torque on the oscillating body is zero.
• Displacement: The instantaneous distance and direction of the oscillator from its equilibrium position.
• Amplitude: The maximum magnitude of displacement from the equilibrium position.
• Time period: The time taken to complete one full oscillation, represented by $T$.
• Frequency: The number of complete oscillations per unit time, represented by $f$, where $f=1/T$.
• Angular frequency: The rate of change of phase, represented by $\omega$, where $\omega =2\pi f=2\pi /T$.
• Phase: A quantity that specifies the state of oscillation at a particular instant.
• Restoring force: A force that acts toward the equilibrium position and tends to bring the body back to it.
• Simple harmonic motion: Oscillatory motion in which acceleration is proportional to displacement and opposite in direction: $a=-{\omega }^{2}x$.
• SHM displacement equation: The displacement of an oscillator can be written as $x=A\sin (\omega t+\varphi )$ or $x=A\cos (\omega t+\varphi )$, where $\varphi$ is the initial phase.
• Velocity in SHM: The velocity is $v=\text{plus or minus }\omega \sqrt{(A^2-x^2)}$; it is maximum at the mean position and zero at the extreme positions.
• Acceleration in SHM: The acceleration is $a=-{\omega }^{2}x$; it is zero at the mean position and maximum in magnitude at the extremes.
• Spring-mass oscillator: A mass attached to a spring performs SHM when the spring obeys Hooke's law, with time period $T=2\pi \sqrt{m/k}$.
• Simple pendulum: A small bob suspended by a light, inextensible string that oscillates about its lowest position.
• Pendulum time period: For small angular oscillations, $T=2\pi \sqrt{l/g}$, independent of the bob's mass and approximately independent of amplitude.
• Mechanical energy in SHM: The total mechanical energy remains constant in an ideal oscillator and equals $E=\frac{1}{2}k{A}^2=\frac{1}{2}m{\omega }^2{A}^2$.
• Damped oscillation: Oscillation whose amplitude gradually decreases because energy is lost through friction or other resistive forces.
• Forced oscillation: Oscillation produced or maintained by an external periodic force.
• Resonance: A condition in which the amplitude becomes very large when the driving frequency is close to the natural frequency of the system.
• Physical pendulum: An extended rigid body oscillating about a horizontal axis.
• Potential energy in a spring oscillator: $U=\frac{1}{2}k{x}^2$.
• Kinetic energy in a spring oscillator: $K=\frac{1}{2}m{v}^2$.
• SI unit of displacement and amplitude: Metre.
• SI unit of time period: Second.
• SI unit of frequency: Hertz.
• SI unit of angular frequency: Radian per second.
• SI unit of energy: Joule.

Easily Confused

• Periodic motion and oscillatory motion: Periodic motion repeats after equal time intervals, whereas oscillatory motion specifically involves to-and-fro motion about an equilibrium position.
• Amplitude and displacement: Displacement is the instantaneous position relative to equilibrium; amplitude is the maximum possible magnitude of displacement.
• Mean position and extreme position: Speed and kinetic energy are maximum at the mean position, while speed is zero and potential energy is maximum at either extreme.
• Simple pendulum and physical pendulum: A simple pendulum is modeled as a point mass on a massless string; a physical pendulum is an extended rigid body oscillating about a horizontal axis.
• Damped and forced oscillation: Damping reduces amplitude through energy loss, whereas forced oscillation is maintained or produced by an external periodic force.
• Forced oscillation and resonance: Forced oscillation is the general response to an external periodic force; resonance is the large-amplitude condition when the driving frequency is close to the natural frequency.
• Frequency and angular frequency: Frequency $f$ is measured in hertz and counts oscillations per second; angular frequency $\omega$ is measured in radians per second and equals $2\pi f$.
• Simple pendulum independence from mass and amplitude: The pendulum period does not depend on bob mass and is approximately independent of amplitude only for small oscillations.

What Gets Asked

• Defining SHM: State that the restoring acceleration is proportional to displacement and opposite in direction, using $a=-{\omega }^{2}x$. Omitting the direction condition or the negative sign loses the defining feature.
• Spring-mass calculations: Use $\omega =\sqrt{k/m}$, $T=2\pi \sqrt{m/k}$, or $f=(1/2\pi )\sqrt{k/m}$. Confusing mass and spring constant gives the wrong dependence.
• Simple pendulum calculations: Apply $T=2\pi \sqrt{l/g}$, remembering that the formula requires small angular amplitudes and does not involve the bob's mass.
• Position-based descriptions: Compare the mean and extreme positions using displacement, speed, acceleration, kinetic energy and potential energy. Stating that speed is maximum at an extreme rather than at equilibrium is a common error.
• Energy in SHM: Explain the exchange between kinetic and potential energy while total mechanical energy remains constant, using $E=K+U=\frac{1}{2}k{A}^2$. Do not state that kinetic and potential energy are both maximum at the same position.
• Real oscillations and resonance: Distinguish damping, forced oscillation and resonance, including the fact that resonance produces very large amplitude when the driving frequency is close to the natural frequency.