# SHM Graphs ## Representing Simple Harmonic Motion Graphically - The displacement, velocity and acceleration of an object in simple harmonic motion can be represented by graphs against time - All undamped SHM graphs are represented by **periodic functions** - This means they can all be described by sine and cosine curves - The motion repeats in a regular pattern over time

Periodic functions in SHM are mathematical functions that repeat their values at regular intervals, producing the characteristic wave patterns of oscillatory motion

## Displacement-Time Graph - The displacement graph shows how far the oscillator is from equilibrium at each moment - **Key features:** - The amplitude $A$ (or $x_0$) can be found from the maximum value of $x$ - The time period $T$ can be found by reading the time taken for one complete cycle - The graph might not always start at 0 - If the oscillations start at the positive or negative amplitude, the displacement will be at its maximum - The equation depends on starting position: - Starting from equilibrium: $x = x_0 \sin \omega t$ - Starting from maximum displacement: $x = x_0 \cos \omega t$ **Example: Displacement-time graph for an oscillator starting from equilibrium position:**

This shows a sine curve with amplitude 2 cm and time period 1.0 s. **Example: Displacement-time graph for an oscillator starting from maximum displacement:**

This shows a cosine curve with the same amplitude and time period. ## Velocity-Time Graph - The velocity graph shows the speed and direction of the oscillator at each instant - **Key features:** - It is **90° out of phase** with the displacement-time graph - Velocity equals the rate of change of displacement - The velocity of an oscillator at any time can be determined from the **gradient of the displacement-time graph:** $$v = \frac{\Delta x}{\Delta t}$$ - An oscillator moves fastest at its equilibrium position - Therefore, velocity is at its maximum when displacement is zero - This occurs when the object passes through $x = 0$ - The maximum velocity is given by: $v_0 = \omega x_0$ - The equation depends on starting position: - Starting from equilibrium: $v = v_0 \cos \omega t$ - Starting from maximum displacement: $v = -\omega x_0 \sin \omega t$

At equilibrium (x = 0), velocity is maximum; at maximum displacement (x = ±x₀), velocity is zero.

**Example: Velocity-time graph for the same oscillator starting from equilibrium:**

Notice that the velocity is maximum at $t = 0$ when displacement is zero, and velocity is zero at $t = 0.25$ s when displacement is maximum. ## Acceleration-Time Graph - The acceleration graph shows the rate of change of velocity at each instant - **Key features:** - The acceleration graph is a **reflection of the displacement graph** on the $x$-axis - This means when a mass has positive displacement (to the right), the acceleration is in the opposite direction (to the left) and vice versa - It is **90° out of phase** with the velocity-time graph - Acceleration equals the rate of change of velocity - The acceleration of an oscillator at any time can be determined from the **gradient of the velocity-time graph:** $$a = \frac{\Delta v}{\Delta t}$$ - The maximum value of acceleration is when the oscillator is at its **maximum displacement** - This occurs at the amplitude positions $x = \pm x_0$ - The maximum acceleration is given by: $a_{\text{max}} = \omega^2 x_0$ **Example: Acceleration-time graph for the same oscillator starting from equilibrium:**

This is a negative sine curve - the mirror image of the displacement graph. Maximum acceleration magnitude occurs at maximum displacement. ## Phase Relationships - The displacement, velocity and acceleration graphs in SHM are all **90° out of phase** with each other - This means: - When displacement is maximum, velocity is zero - When velocity is maximum, acceleration is zero - When acceleration is maximum, displacement is maximum (but in opposite direction) - The graphs form a sequence where each one leads or lags the next by a quarter cycle

The equations differ depending on the starting point of the oscillator at t = 0 — whether it begins at the equilibrium position or at maximum displacement.

**Comparing all three graphs together shows the phase relationships clearly:**

Note: The velocity and acceleration values have been scaled down for clearer comparison. The key observation is the phase relationship - each curve is shifted by one quarter of a period (0.25 s) relative to the next. ## Summary of Equations and Graphs | Quantity | Object starts in centre of motion | Object starts at extremes of motion | |----------|-----------------------------------|-------------------------------------| | **Displacement** | $x = x_0 \sin \omega t$ | $x = x_0 \cos \omega t$ | | | Graph: sine curve starting at origin | Graph: cosine curve starting at maximum | | **Velocity** | $v = \omega x_0 \cos \omega t$ | $v = -\omega x_0 \sin \omega t$ | | | $v = \pm \omega\sqrt{x_0^2 - x^2}$ | $v = \pm \omega\sqrt{x_0^2 - x^2}$ | | | Graph: cosine curve (90° ahead of displacement) | Graph: negative sine curve | | **Acceleration** | $a = -x_0\omega^2 \sin \omega t$ | $a = -x_0\omega^2 \cos \omega t$ | | | $a = -\omega^2 x$ | $a = -\omega^2 x$ | | | Graph: negative sine curve (90° ahead of velocity) | Graph: negative cosine curve | **Important relationships:** - Maximum velocity: $v_{\text{max}} = \omega x_0$ (occurs at $x = 0$) - Maximum acceleration: $a_{\text{max}} = \omega^2 x_0$ (occurs at $x = \pm x_0$) - Angular frequency: $\omega = \frac{2\pi}{T} = 2\pi f$ ## Worked Example **Question:** A child's toy bounces vertically with simple harmonic motion. It is released from rest at a height 8 cm above its equilibrium position. The variation of vertical displacement $x$ of the toy with time $t$ is shown below:

Displacement / cm	Time / s
8	0.1
6	0.2
4	0.3
2	0.4
0	0.5
-2	0.6
-4	0.7
-6	0.8
-8	0.9

The toy exhibits simple harmonic motion. Use data from the graph to determine at what time the speed of the toy is first at its maximum. **Answer:** - The velocity is at its maximum when the displacement $x = 0$ - This is because maximum speed occurs at the equilibrium position - Reading the value of time when $x = 0$ from the graph, this equals **0.5 s** - Therefore, at **0.5 s**, the toy will reach its maximum velocity for the first time **Key point:** Maximum velocity always occurs when the object passes through equilibrium, regardless of which direction it's moving

Remember: your calculator must be in radians mode when using sine and cosine functions, since angular frequency ω is in rad s⁻¹

## Summary of Key Concepts - **Displacement graph:** Shows position relative to equilibrium - Maximum at amplitude, zero at equilibrium - Sine or cosine curve depending on starting position - **Velocity graph:** Shows rate of change of displacement - Maximum at equilibrium, zero at amplitude - 90° ahead of displacement graph - **Acceleration graph:** Shows rate of change of velocity - Maximum at amplitude, zero at equilibrium - Reflection of displacement graph (opposite direction) - 90° ahead of velocity graph - All three quantities vary continuously and periodically - The time period $T$ is the same for all three graphs - The shape depends on whether oscillations start from equilibrium or amplitude