Topic 7.1 Defining Simple Harmonic Motion: identify simple harmonic motion as arising from a linear restoring force, derive the defining differential equation, and recognize its sinusoidal solution.

What this topic is asking

The College Board (Topic 7.1) wants you to identify simple harmonic motion (SHM) as the motion produced by a linear restoring force, to derive the defining differential equation, and to recognize its sinusoidal solution. This is the calculus core of Unit 7: SHM is defined by a second-order differential equation, and everything else in the unit, frequency, energy, pendulums, flows from it.

The linear restoring force

The defining ingredient is the linear restoring force. Whenever an object is displaced from a stable equilibrium and the force pulling it back grows in proportion to the displacement, the motion is SHM. The spring is the prototype (Hooke's law $F = -kx$), but the same form appears for a pendulum at small angles and for many systems near a potential-energy minimum. Because the force is restoring, the system overshoots equilibrium, swings to the other side, and oscillates indefinitely (in the absence of damping).

The defining differential equation

Applying Newton's second law to the linear restoring force gives a second-order differential equation:

where we have defined $\omega^2 = k/m$. This equation, the acceleration is proportional to the negative of the displacement, is the mathematical definition of SHM. Any system you can reduce to this form oscillates harmonically, and identifying the coefficient gives you the angular frequency immediately. Deriving this equation for a given system, a spring, a floating object, a small-angle pendulum, is the central AP Physics C skill of the unit.

The sinusoidal solution

The solution to $\ddot{x} = -\omega^2 x$ is sinusoidal:

where $A$ is the amplitude (maximum displacement) and $\phi$ is the phase constant, both set by the initial position and velocity. You can verify it by differentiating twice: $\ddot{x} = -\omega^2 A\cos(\omega t + \phi) = -\omega^2 x$, which matches. The key feature is that the angular frequency $\omega = \sqrt{k/m}$ depends only on the stiffness and mass, not on the amplitude. This amplitude-independence (isochronism) means a mass-spring oscillator keeps the same period whether it swings far or near, a property that makes harmonic oscillators useful as clocks.

Try this

Q1. A mass-spring system has $\ddot{x} = -64x$ (SI units). State its angular frequency. [2 points]

• Cue. $\omega^2 = 64$, so $\omega = 8.0$ rad/s.

Q2. Explain why the period of a mass-spring oscillator does not depend on the amplitude. [2 points]

• Cue. The angular frequency $\omega = \sqrt{k/m}$ depends only on $k$ and $m$, so the period is fixed regardless of how far the mass is displaced.