IB Physics: Simple Harmonic Motion

Simple Harmonic Motion (SHM) is the foundational model for understanding oscillations, from a child on a swing to the atoms in a crystal lattice. Mastering SHM is crucial for the IB Physics syllabus because it connects core principles of kinematics, dynamics, and energy conservation into a single, elegant framework. Your ability to derive its equations, interpret its graphs, and apply it to systems like springs and pendulums is essential for both internal assessments and final exams.

Defining Simple Harmonic Motion

An object undergoes Simple Harmonic Motion (SHM) when its acceleration is directly proportional to its displacement from a fixed equilibrium point and is always directed towards that point. This definition is captured by the defining equation:

$$a=-{\omega }^{2}x$$

Here, $a$ is acceleration, $x$ is displacement from equilibrium, and $\omega$ (omega) is the angular frequency. The negative sign is the critical component, indicating the restoring nature of the force; acceleration always points back to the center.

The key condition for SHM is this linear, restoring relationship. Not all oscillations are simple harmonic. A ball bouncing on the ground, for instance, experiences a constant gravitational acceleration, not one proportional to displacement, so it is not SHM. Common systems that do exhibit SHM include an ideal mass-spring system on a frictionless surface and a simple pendulum for small angular displacements (typically less than 10°).

The Kinematics of SHM: Displacement, Velocity, and Acceleration

From the defining equation $a=-{\omega }^{2}x$, we can derive the equations for displacement, velocity, and acceleration as functions of time. The solutions are sinusoidal, reflecting the periodic, back-and-forth motion. Starting from a maximum displacement $x_0$ (the amplitude), the equations are:

• Displacement: $x=x_0\cos (\omega t+\varphi )$
• Velocity: $v=-\omega x_0\sin (\omega t+\varphi )$
• Acceleration: $a=-{\omega }^2{x}_0\cos (\omega t+\varphi )$

In these equations, $\varphi$ (phi) is the phase constant, which determines the starting point of the oscillation. The term $(\omega t+\varphi )$ is called the phase. The time for one complete cycle is the period (T), related to angular frequency by $T=\frac{2\pi }{\omega }$. The frequency (f), measured in hertz (Hz), is the number of cycles per second: $f=\frac{1}{T}=\frac{\omega }{2\pi }$.

A crucial skill for IB exams is interpreting and sketching the graphs of these three functions. They are all sinusoidal but out of phase. The velocity graph is the gradient (derivative) of the displacement graph, and the acceleration graph is the gradient of the velocity graph. When displacement is at a maximum (amplitude), velocity is zero, and acceleration is at a maximum in the negative direction.

Energy Transformations in an SHM System

A system undergoing undamped SHM is conservative; the total mechanical energy remains constant. This energy continuously transforms between kinetic energy (KE) and potential energy (PE). For a mass-spring system, the potential energy is elastic ($E_{el}$). For a pendulum, it is gravitational ($E_g$).

The total energy $E_T$ in the system is constant and equal to the maximum potential energy (at maximum displacement, where KE=0) or the maximum kinetic energy (at equilibrium, where PE=0).

For a mass-spring system: $$E_T=\frac{1}{2}k{x}_0^2=\frac{1}{2}m{v}_{max}^2$$

Here, $k$ is the spring constant and $v_{max}=\omega x_0$ is the maximum velocity. At any point, the energy is partitioned as: $$E_T=\frac{1}{2}k{x}^2+\frac{1}{2}m{v}^2$$

This energy relationship provides a powerful problem-solving tool. For example, you can often find velocity at a given displacement more easily using energy conservation ($v=\pm \omega \sqrt{x_0^2-x^2}$) than by using the velocity-time equation directly.

Damped, Forced Oscillations and Resonance

Real-world oscillations are seldom perfectly simple harmonic due to damping—the loss of energy from the system, usually to friction or air resistance. Damping reduces the amplitude over time. Light damping causes a gradual amplitude decrease, while critical damping returns the system to equilibrium in the shortest possible time without oscillating. Overdamping does so more slowly.

When an external periodic driving force is applied to an oscillating system, it undergoes forced oscillations. The system's response depends dramatically on the relationship between the driving frequency ($f_d$) and its natural frequency of oscillation ($f_0$).

Resonance is the dramatic increase in amplitude that occurs when the driving frequency matches the natural frequency of the system ($f_d=f_0$). At resonance, energy is transferred most efficiently from the driving force to the oscillator. This has profound applications and dangers: it allows for the tuning of a radio circuit (desirable) but can also cause bridges to collapse in high winds or buildings to sway during earthquakes (catastrophic).

Applying SHM: Mass-Spring Systems and the Simple Pendulum

Two cornerstone applications of SHM in IB Physics are the mass-spring system and the simple pendulum.

1. Mass-Spring System: For a horizontal spring with mass $m$ and spring constant $k$, the restoring force is given by Hooke's Law, $F=-kx$. Applying Newton's Second Law ($F=ma$) yields $a=-\frac{k}{m}x$. Comparing this to the SHM definition $a=-{\omega }^{2}x$ gives the angular frequency for a spring: $$\omega =\sqrt{\frac{k}{m}}$$ Therefore, the period is: $$T=2\pi \sqrt{\frac{m}{k}}$$

2. Simple Pendulum: For a small bob of mass $m$ on a string of length $L$, the restoring force component is $F=-mg\sin \theta$. For small angles (in radians), $\sin \theta \approx \theta$. The displacement arc length is $s=L\theta$, leading to $a=-\frac{g}{L}s$. This fits the SHM form, giving angular frequency: $$\omega =\sqrt{\frac{g}{L}}$$ And period: $$T=2\pi \sqrt{\frac{L}{g}}$$ Crucially, the period of a simple pendulum is independent of the mass of the bob and the amplitude (for small angles).

Common Pitfalls

1. Misapplying the Pendulum Formula: Using $T=2\pi \sqrt{L/g}$ for large amplitudes (e.g., 30°) is incorrect. This formula is an approximation valid only for small angular displacements (less than about 10°). For larger angles, the motion is periodic but not simple harmonic, and the period becomes amplitude-dependent.

1. Confusing Maximum and Instantaneous Values: Students often mistakenly set kinetic energy formulas like $\frac{1}{2}m{v}^2$ equal to potential energy formulas like $\frac{1}{2}k{x}^2$. This is only true if the $v$ and $x$ are the maximum values, or if you are using the total energy equation $\frac{1}{2}m{v}^2+\frac{1}{2}k{x}^2=\frac{1}{2}k{x}_0^2$. At any other point in the cycle, $v$ and $x$ are instantaneous values that do not satisfy the simple equality.

1. Sign Errors with the SHM Equation: Forgetting the negative sign in $a=-{\omega }^{2}x$ or $F=-kx$ is a critical error. The sign defines the motion as restorative. Omitting it implies the acceleration helps increase the displacement, leading to exponential runaway motion, which is the opposite of oscillation.

1. Misinterpreting Phase on Graphs: When asked for the phase difference between displacement and velocity on an exam, a common wrong answer is $\pi /2$ (90°). While correct in magnitude, the full answer is that velocity leads displacement by $\pi /2$ radians, or displacement lags velocity by $\pi /2$ radians. Stating the direction of the phase relationship demonstrates deeper understanding.

Summary

• SHM is defined by the condition $a=-{\omega }^{2}x$, where acceleration is proportional and oppositely directed to displacement.
• The kinematics are described by sinusoidal equations for $x$, $v$, and $a$, which are phase-shifted relative to each other. Maximum displacement corresponds to zero velocity and maximum acceleration.
• Energy is conserved in undamped SHM, constantly transforming between kinetic and potential forms, with the total energy given by $E_T=\frac{1}{2}k{x}_0^2$ for a spring system.
• Resonance occurs in forced oscillations when the driving frequency equals the natural frequency, leading to maximal energy transfer and amplitude.
• Key system periods are: Mass-Spring: $T=2\pi \sqrt{m/k}$; Simple Pendulum (small angle): $T=2\pi \sqrt{L/g}$.