Revision summary

State equilibrium, displacement sign, amplitude and the chosen time origin. Then check whether the motion is ideal free SHM, damped, or maintained by a periodic drive.

Describe a complete cycle

Amplitude x₀ is maximum |x|, not the extreme-to-extreme distance. A full cycle returns to the same state of motion; opposite-direction centre crossings are half a period apart. Frequency counts cycles per second, while angular frequency gives phase advance in rad/s.

Phase identifies the stage of a cycle. For same-frequency traces, a positive time separation Δt between corresponding stages gives phase-difference magnitude 2πΔt/T. Name the leading trace. A quarter-cycle difference is π/2; a half-cycle difference is π. Equal displacement alone does not identify equal phase.

For measurements, calibrate position and timing, mark equilibrium, track one point and time repeated complete cycles. Check rate/amplitude changes and keep the release and apparatus conditions controlled. Repeats do not remove a wrong reference, miscount or perspective error.

Use the restoring condition

SHM needs both proportionality to displacement and acceleration towards equilibrium. Its a-x graph is a straight line through the origin with negative gradient -ω². The sine solution shown chooses t = 0 at a positive-direction centre crossing; a release at the positive extreme needs a different phase.

Select velocity sign and match graph times

Choose the square-root sign from the direction of motion. At equilibrium speed is greatest and restoring acceleration is zero. At either extreme velocity is zero and acceleration is greatest in magnitude towards equilibrium. Velocity leads sine displacement by π/2; acceleration and displacement are in antiphase.

Velocity is the x-t gradient and acceleration the v-t gradient. For sampled positions, Δx/Δt gives interval averages, approximately associated with their midpoints. Pair acceleration estimates with displacement at corresponding times. Use actual calibrated records on both sides of equilibrium to investigate a-x proportionality; generated model data are not independent confirmation.

Smaller intervals reduce a finite-gradient approximation error only while position and timing resolution remain adequate. Differencing amplifies noise. Distinguish approximate interval results from exact instantaneous model values.

Keep the energy account and reference clear

With no losses or drive, total mechanical energy is constant. In the equilibrium-zero reference, it is all kinetic at the centre and all elastic potential at either extreme. Both stores are present between them. Gravitational energy is unchanged for the horizontal model; a vertical model must include its gravitational contribution.

For unchanged mass and linear stiffness, doubling amplitude doubles maximum speed and acceleration magnitude, quadruples energy and leaves period unchanged within the model. Energy repeats every T/2, although the full signed motion repeats after T.

Compare like initial conditions

For the same mass, stiffness, initial displacement and release from rest: light/underdamping gives decreasing oscillations; critical damping gives the fastest nonoscillatory return; heavy/overdamping returns more slowly. These statements refer to the stated comparison, not every possible initial velocity.

Damping transfers mechanical energy to other forms. Car suspension damping limits repeated bouncing, with rapid settling as a design aim. More damping is not always better.

Read the response axes and drive conditions

Natural frequency describes free oscillation; the external driver sets driving frequency. After transients settle, a linear forced response repeats at that driving frequency. Steady amplitude with damping requires continuing energy input to replace losses.

An amplitude-versus-driving-frequency graph peaks at resonance, close to or at the natural frequency under the relevant conditions. At the same drive strength, greater damping lowers and broadens the displacement-response peak; its position need not be exactly the undamped natural frequency.

Resonance can help build a swing's motion or a musical resonator's response, but can cause unwanted machine-support vibration. Identify the driver, responding system and energy transfer. Control the driving-force amplitude when comparing the supplied response curves.

The phase angle φ here is not gravitational potential: the named quantity and its units distinguish those uses. Keep radians in ωt when ω is in rad/s.