Oscillations

The sign does not specify the direction of velocity. At a positive displacement, the oscillator may be travelling farther outward or returning inward, but its acceleration still points back towards equilibrium. Both proportionality and direction are required for SHM; a restoring force alone may be nonlinear and produce another type of oscillation. Plotting acceleration against displacement gives a straight line through the origin with negative gradient. The slope determines ω², allowing period to be inferred without directly timing every oscillation. Label the equilibrium position carefully because displacement must be measured from that point.

At an extreme, the object reverses direction: its velocity is momentarily zero, but the restoring force is largest. At equilibrium, restoring force and acceleration vanish instantaneously, yet the object passes through at its greatest speed because kinetic energy is maximum. These observations do not conflict; velocity describes motion now, while acceleration describes how velocity changes. For a mass on a vertical spring, use displacement from the loaded equilibrium, not extension from the spring’s natural length. The weight shifts equilibrium but does not change the ideal linear oscillation equation about it.

The driving frequency determines the steady forced oscillation frequency, while the amplitude depends on the relationship between driving and natural frequencies. Energy does not accumulate without limit in a realistic steady state because damping removes it. A swing can be driven effectively with appropriately timed pushes; structures may need damping to avoid damaging motion. Distinguish the natural frequency of the undriven system from the exact frequency of maximum displacement amplitude in a damped system. The simple equality is a useful light-damping approximation, rather than an exact description of every resonance curve.

The graph must have acceleration vertically and displacement horizontally for this direct interpretation. A positive gradient would indicate acceleration away from the origin rather than a stable restoring motion under that sign convention. Check displacement units: centimetres on the axis must be converted before reporting a slope in s⁻². A straight line through the origin supports the proportional restoring model, while curvature can show that the motion is outside the linear range. The period inferred this way describes the modelled motion, so compare it with a direct timing measurement as a useful consistency check.

The period depends on m and k through 2π√(m/k). A larger amplitude means a larger restoring force and greater acceleration, which allow the object to cover the longer path in the same ideal time. That reasoning fails if spring stiffness changes with extension or damping is significant. For a pendulum, the familiar amplitude-independent period is a small-angle approximation; large swings take longer. Keep these assumptions explicit when comparing measurements, because an observed period change may reveal a model limitation rather than a mistake in timing.

Starting and stopping on opposite-direction crossings can introduce a half-period counting error. Timing many cycles reduces the relative contribution of reaction-time uncertainty, but repeat trials are still useful for estimating scatter. The model treats the bob as a point mass and the string as light and effectively inextensible. Measuring only the string length underestimates the pendulum length if the bob radius is appreciable. Plot T² against L for a linear test, and consider whether a nonzero intercept could come from a consistent length offset rather than forcing every line through the origin.