SHM Graphical Analysis and Energy Transfers

Simple harmonic motion (SHM) is far more than a mathematical abstraction; it is the fundamental model for oscillations from the swing of a pendulum to the vibration of atoms in a crystal. Mastering its graphical and energetic analysis transforms it from a set of equations into an intuitive understanding of how systems store and exchange energy over time. This knowledge is crucial for tackling A-Level problems that seamlessly blend kinematics, dynamics, and conservation laws.

Defining SHM and Its Core Equations

Simple harmonic motion is defined as oscillation where the acceleration of the object is directly proportional to its displacement from a fixed equilibrium point, and is always directed towards that point. This is captured by the defining equation: $a=-{\omega }^{2}x$, where $a$ is acceleration, $x$ is displacement, and $\omega$ is the angular frequency (measured in rad s⁻¹).

From this, we derive the key kinematic equations that describe the motion at any time $t$:

• Displacement: $x=A\cos (\omega t+\varphi )$
• Velocity: $v=-\omega A\sin (\omega t+\varphi )$
• Acceleration: $a=-{\omega }^{2}A\cos (\omega t+\varphi )$

Here, $A$ is the amplitude (maximum displacement) and $\varphi$ is the phase constant, which depends on initial conditions. The maximum values are critical: maximum velocity $v_{max}=\omega A$ and maximum acceleration $a_{max}={\omega }^{2}A$. For a spring-mass system, $\omega =\sqrt{k/m}$; for a simple pendulum (with small angles), $\omega =\sqrt{g/L}$.

Phase Relationships and Sketching Graphs

The sine and cosine forms of the equations reveal consistent phase differences, which are clearly visible when sketching displacement-time ($x$-$t$), velocity-time ($v$-$t$), and acceleration-time ($a$-$t$) graphs. Assuming $\varphi =0$ (starting from maximum displacement), displacement is a cosine curve.

Velocity leads displacement by a phase of $\pi /2$ (or 90°). When displacement is at a maximum, velocity is zero. As the object passes through equilibrium, displacement is zero but velocity is at its maximum magnitude. Graphically, the $v$-$t$ graph is a negative sine curve, which is the derivative (gradient) of the $x$-$t$ graph.

Acceleration is in anti-phase with displacement (a phase difference of $\pi$ or 180°). Acceleration is always directed towards equilibrium, so when displacement is positive maximum, acceleration is negative maximum. The $a$-$t$ graph is an inverted cosine curve. Importantly, the acceleration graph is the derivative of the velocity graph.

Example: For an object released from rest at $x=+A$, sketch the graphs over one period $T$. The $x$-$t$ graph starts at $A$ as a cosine. The $v$-$t$ graph starts at 0 as a negative sine. The $a$-$t$ graph starts at $-a_{max}$ as a negative cosine.

The Velocity-Displacement Ellipse and Energy Graphs

Plotting velocity against displacement yields one of the most insightful graphs in SHM. Squaring and combining the $x$ and $v$ equations eliminates time ($t$) to give: $$v^2={\omega }^2(A^2-x^2)$$. Rearranged, this becomes $$\frac{x^2}{A^2}+\frac{v^2}{(\omega A{)}^2}=1$$, which is the equation of an ellipse. The ellipse visually encapsulates energy conservation: at $x=0$, $v$ is maximum ($\pm \omega A$); at $x=\pm A$, $v=0$.

This leads directly to energy analysis. In a spring-mass system, the total mechanical energy is constant and equals the maximum elastic potential energy: $E_{total}=\frac{1}{2}k{A}^2$.

• Elastic Potential Energy (EPE): $E_p=\frac{1}{2}k{x}^2$. This plots as a parabola ($y\propto x^2$) centred on $x=0$.
• Kinetic Energy (KE): $E_k=\frac{1}{2}m{v}^2=\frac{1}{2}k(A^2-x^2)$, from the ellipse equation. This plots as an inverted parabola.

On an energy-displacement graph, the parabolic $E_p$ curve and the inverted parabolic $E_k$ curve sum vertically to a constant horizontal line representing $E_{total}$. The point where the $E_k$ and $E_p$ curves intersect is where $E_k=E_p$, which occurs at $x=\pm A/\sqrt{2}$.

Solving Problems with Energy Conservation

Energy conservation provides a powerful, often simpler, method for solving SHM problems without resolving trigonometric functions. The fundamental principle is: (Total Energy) = (Kinetic Energy) + (Potential Energy) = constant.

Worked Example (Spring-Mass): A 2.0 kg mass oscillates on a spring ($k=200$ N m⁻¹) with amplitude 0.10 m. Find (a) the maximum speed, (b) the speed when displacement is 0.05 m, and (c) the potential energy when speed is 2.0 m s⁻¹.

Step 1: Find constants. Total energy: $E_{total}=\frac{1}{2}k{A}^2=\frac{1}{2}\ast 200\ast (0.10{)}^2=1.0$ J. Angular frequency: $\omega =\sqrt{k/m}=\sqrt{200/2.0}=10$ rad s⁻¹.

Step 2: Solve. (a) Maximum speed: $v_{max}=\omega A=10\ast 0.10=1.0$ m s⁻¹. (Check via energy: $\frac{1}{2}m{v}_{max}^2=1.0$ J, gives same result). (b) Use $v^2={\omega }^2(A^2-x^2)$: $v^2=10^2\ast (0.10^2-0.05^2)=100\ast (0.01-0.0025)=0.75$. Thus $v=\sqrt{0.75}\approx 0.87$ m s⁻¹. (c) Use $E_{total}=E_k+E_p$. $E_k=\frac{1}{2}\ast 2.0\ast (2.0{)}^2=4.0$ J. This is immediately a problem—4.0 J > 1.0 J total energy. Therefore, a speed of 2.0 m s⁻¹ is not possible for this oscillator, as it exceeds $v_{max}$. This highlights the usefulness of energy as a sanity check.

For a simple pendulum, the potential energy is gravitational ($mgh$), but for small angles, the motion approximates SHM with energy interchanging between kinetic and gravitational potential.

Common Pitfalls

1. Misinterpreting Graph Gradients: A common error is thinking the gradient of an $x$-$t$ graph gives acceleration. Remember, gradient of $x$-$t$ = velocity; gradient of $v$-$t$ = acceleration. Confusing this leads to incorrect phase analysis.

Correction: Practice sketching all three graphs for the same motion, noting that each is the derivative of the previous.

1. Confusing Maximum Values: Students often misremember that maximum velocity occurs at maximum displacement. The defining equation $a=-{\omega }^{2}x$ shows acceleration (not velocity) is maximum at maximum displacement. Velocity is maximum at equilibrium ($x=0$).

Correction: Link maximums to energy: PE max at extremes (v=0), KE max at centre (v max).

1. Misapplying the Ellipse Equation: When using $v^2={\omega }^2(A^2-x^2)$, forgetting that $v$ can be positive or negative. The equation gives $v^2$, so you must take the square root and consider the direction of motion from context.

Correction: Solve for $v^2$ first, then take $\pm \sqrt{}$ and decide the sign based on whether the object is moving towards or away from equilibrium.

1. Neglecting the SHM Conditions for a Pendulum: Treating all pendulum motions with SHM equations. The SHM equations for a pendulum ($\omega =\sqrt{g/L}$) are only valid for small angular amplitudes (typically < 10°). For larger angles, the motion is periodic but not SHM, and the period becomes amplitude-dependent.

Correction: Always check if a problem states "small angle" or gives an amplitude in degrees. If not, you cannot blindly use the SHM formulas.

Summary

• The defining condition for SHM is $a=-{\omega }^{2}x$, leading to sinusoidal equations for displacement, velocity, and acceleration with fixed phase relationships: velocity leads displacement by 90°, and acceleration is in anti-phase with displacement.
• Plotting velocity against displacement produces an ellipse described by $v^2={\omega }^2(A^2-x^2)$, a direct consequence of energy conservation.
• Energy in SHM is shared between kinetic and potential forms. On an energy-displacement graph, kinetic energy and potential energy are parabolic curves that sum to a constant total energy line.
• Maximum values are key: $v_{max}=\omega A$ and $a_{max}={\omega }^{2}A$. These are derived from and consistent with energy conservation: $E_{total}=\frac{1}{2}m{v}_{max}^2=\frac{1}{2}k{A}^2$.
• Energy conservation provides a robust problem-solving framework for both spring-mass systems and simple pendulums (under small-angle conditions), often simplifying calculations compared to using trigonometric functions directly.