Periodic motion (AQA A-Level Physics): Revision Notes

Simple harmonic motion (SHM)

Introduction to oscillations and periodic motion

Oscillations and vibrations represent a particular type of periodic motion, where the motion repeats itself in a consistent, regular pattern as time progresses. When an object oscillates, it repeatedly moves back and forth around a central equilibrium position.

A mechanical oscillation differs from oscillations in fields (such as electromagnetic waves) because it requires a physical restoring force. This restoring force always acts toward the equilibrium position. The distance that an oscillating object moves from its equilibrium position is called the displacement, represented by the symbol $x$.

Defining simple harmonic motion

Simple harmonic motion (SHM) represents a special category of oscillatory motion. The defining characteristic of SHM is:

A mass oscillating on a spring provides a classic example of SHM. This motion can be monitored using data logging equipment connected to a motion sensor. When the displacement versus time data is analyzed by computer, it produces either a sine curve or a cosine curve, depending on the starting conditions.

Key characteristics of SHM

Amplitude

The amplitude (symbol: $A$) represents the maximum displacement from the equilibrium position. It measures how far the oscillating object travels from its central position to either extreme.

Time period and cycles

The time period (symbol: $T$) is the time taken for one complete cycle of oscillation. A cycle corresponds to the oscillating mass moving through one position, continuing through its motion, and then passing through that same position again while moving in the same direction.

Frequency and angular frequency

The frequency (symbol: $f$) represents the number of complete cycles of oscillation that occur per second. It is measured in hertz (Hz), where $1 \text{ Hz} = 1 \text{ s}^{-1}$. The frequency relates to the time period through the equation:

$f = \frac{1}{T}$

This means that as the time period increases, the frequency decreases, and vice versa.

The angular frequency (symbol: $\omega$) is measured in $\text{rad} \cdot \text{s}^{-1}$ and provides an alternative way to characterize the oscillation. It relates to the time period and frequency through:

$\omega = \frac{2\pi}{T}$

Since frequency equals $1/T$, we can also write:

$\omega = 2\pi f$

The angular frequency represents how rapidly the oscillation progresses through its cycle when expressed in radians.

Equations of SHM

Displacement equations

The displacement of an object undergoing SHM varies sinusoidally with time. The specific form of the equation depends on the initial conditions when monitoring begins.

If the object is at maximum displacement (at an extreme position) when timing starts at $t = 0$, the displacement follows:

$x = A\cos(\omega t)$

where $x$ is the displacement at time $t$, $A$ is the amplitude, and $\omega$ is the angular frequency.

Alternatively, if the oscillating object passes through its equilibrium position (where $x = 0$) when timing starts at $t = 0$, the displacement follows:

$x = A\sin(\omega t)$

Acceleration in SHM

The defining equation that characterizes simple harmonic motion mathematically is:

When displacement is positive (object on one side of equilibrium), acceleration is negative (pointing back toward equilibrium), and vice versa.

The maximum value for acceleration occurs when the displacement reaches its maximum value (the amplitude $A$):

$a_{\text{max}} = \omega^2 A$

Since acceleration is directly proportional to the resultant force (from Newton's second law, $F = ma$), and SHM requires a restoring force directly proportional to displacement, we can see how this equation captures the essential nature of SHM. A graph plotting acceleration versus displacement for an object in SHM produces a straight line passing through the origin with a gradient of $-\omega^2$. For a restoring force versus displacement graph, the line would pass through the origin with gradient $-m\omega^2$, where $m$ is the mass of the oscillating object.

Velocity in SHM

The velocity of an object moving with SHM varies according to:

$v = \pm 2\pi f\sqrt{A^2 - x^2}$

or equivalently:

$v = \pm \omega\sqrt{A^2 - x^2}$

This equation shows that velocity depends on both the amplitude and the current displacement. The $\pm$ symbol indicates that velocity can be positive or negative depending on the direction of motion.

The maximum velocity occurs when the object passes through the equilibrium position, where $x = 0$. At this point:

$v_{\text{max}} = 2\pi f A = \omega A$

Phase relationships in SHM

The three quantities—displacement, velocity, and acceleration—vary with time in related but shifted patterns. These shifts can be described in terms of phase differences.

Phase difference between displacement and velocity

When examining graphs of displacement and velocity versus time, a phase difference of one quarter of a cycle exists between them. This corresponds to $90°$ or $\pi/2$ radians.

Specifically, the velocity reaches its maximum value when displacement equals zero (at equilibrium), and velocity equals zero when displacement reaches its maximum value (at the extremes). This quarter-cycle shift means that when displacement is at a peak, velocity is crossing through zero; when displacement is at zero, velocity is at a peak.

Phase difference between displacement and acceleration

The phase difference between displacement and acceleration equals half of a cycle, corresponding to $180°$ or $\pi$ radians. This means they reach their maximum values at the same instant but in opposite directions.

Graphical representations of SHM

Displacement-time graphs

A displacement versus time graph for SHM produces a smooth sinusoidal curve (either sine or cosine depending on initial conditions). The amplitude can be read directly from the graph as the maximum displacement from the horizontal axis. The time period can be determined by measuring the time for one complete cycle.

The gradient (slope) of the displacement-time graph at any point gives the velocity at that instant. The gradient is zero when displacement is at a maximum (at the turning points), and the gradient reaches its maximum magnitude when the displacement is zero (at equilibrium).

Velocity-time graphs

Since velocity is the gradient of the displacement-time graph, a velocity versus time graph can be obtained from the displacement graph. For SHM, this also produces a sinusoidal curve, but shifted by a quarter cycle relative to the displacement graph.

The velocity oscillates between $+v_{\text{max}}$ and $-v_{\text{max}}$, passing through zero twice per cycle (at the instants when displacement is maximum). The gradient of the velocity-time graph gives the acceleration at each instant.

Acceleration-time graphs

The acceleration versus time graph, obtained from the gradient of the velocity-time graph, also follows a sinusoidal pattern. This graph has the same shape as the displacement-time graph but inverted (flipped upside down), reflecting the negative sign in the equation $a = -\omega^2 x$. The acceleration oscillates between $+a_{\text{max}}$ and $-a_{\text{max}}$.

Acceleration-displacement graphs

When acceleration is plotted against displacement (rather than against time), the result is a straight line passing through the origin. The gradient of this line equals $-\omega^2$. This linear relationship graphically demonstrates the defining characteristic of SHM: acceleration is directly proportional to displacement but in the opposite direction.

Worked example: mass-spring system

Summary