AP Physics C: Mechanics · Unit 7: Oscillations · Lesson 7.3

Deep Dive: Representing and Analyzing SHM

🔬 Deep Dive

This is your textbook for this topic. Take your time. Read it more than once.

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Representing SHM with Graphs

For an object exhibiting SHM, displacement measured from equilibrium can be written as a sine or cosine function of time:

x = A cos(2πft)  or  x = A sin(2πft)

Velocity and acceleration turn out to be sinusoidal too — and all three graphs are tightly linked. The minima, maxima, and zeros of displacement, velocity, and acceleration are exactly the features you use to read an SHM graph at a glance.

Scrub through one full cycle and watch how displacement, velocity, and acceleration line up — exactly like the three stacked graphs in this unit's header art.

t / T0.30

x/A = -0.31, v/(ωA) = -0.95, a/(ω²A) = 0.31Between the extreme and equilibrium — all three are somewhere in between.

🔑Notice the pattern: whenever displacement is at an extreme (x = ±A), velocity is zero and acceleration is at its own extreme. Whenever displacement is zero (equilibrium), velocity is at its extreme and acceleration is zero. Velocity and acceleration are never at their extremes at the same time as each other.

These properties of SHM — amplitude, timing of extrema, zero crossings — can all be read directly off a graph without ever writing down an equation. That graphical fluency is exactly what the AP exam's Translation Between Representations question tests.

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The SHM Differential Equation

Applying Newton's second law to a restoring force F = −kΔx produces a second-order differential equation — and the position function for SHM is exactly its solution:

d²x/dt² = −ω²x

💡AP Physics C: Mechanics only expects you to know and use the solution to this equation, and to be able to identify SHM from it — not to prove mathematically that the solution is correct.

The solution is the general SHM position function, and everything else in this section falls directly out of it:

x = A cos(ωt + φ)

Since acceleration is the second derivative of position, and the equation above already tells you what that second derivative equals:

a = −ω²x

And the maximum values of velocity and acceleration — the peaks of their sinusoidal graphs — are set entirely by amplitude and angular frequency:

vmax = ωA

amax = ω²A

Adjust amplitude and angular frequency to see how they set the maximum speed and maximum acceleration of an SHM system — then check a = −ω²x at any displacement.

A (m)0.5

ω (rad/s)4

vmax = ωA = 2.00 m/samax = ω²A = 8.00 m/s²

check x (m)0.50

a = −ω²x = -8.00 m/s²

⚠️Changing a system's amplitude changes vmax and amax — but it does not change the period. A pendulum swung through a wider arc or a spring stretched farther still completes each cycle in the same amount of time (for ideal SHM); amplitude and period are independent of each other.

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Resonance

Every oscillating system has a natural frequency — the frequency it oscillates at when displaced from equilibrium and left alone. But what happens when an outside, sinusoidally varying force pushes on the system too?

🔑Resonance occurs when an external force is exerted at exactly a system's natural frequency. When that happens, resonance dramatically increases the amplitude of the oscillating motion — often far beyond what the same-strength force would produce at any other driving frequency.

This system has a natural frequency ω0 = 5 rad/s. Slide the driving frequency ω and watch the response amplitude spike as it approaches ω0 — the same resonance curve shape shown in this unit's header art. (The exact amplitude formula here goes beyond what the AP exam tests — the exam only expects the qualitative idea below.)

driving ω (rad/s)3.0

response amplitude at ω = 3.0: 31% of the peak

This is exactly the driven-oscillator apparatus and resonance curve shown in this unit's header art: as the driving frequency ω sweeps toward the system's natural frequency ω0, the response amplitude climbs to a sharp peak, then falls back off on the other side.

💡Resonance shows up everywhere — a child pumping a swing at just the right rhythm, a wine glass shattering at a matching sung note, a bridge deck responding to wind gusts, a radio tuner picking one station's frequency out of the air. In every case, the trigger is the same: driving force frequency matches natural frequency.

← Back to Lesson 7.3Ready for 7.4? Energy of Simple Harmonic Oscillators tracks where all that kinetic and potential energy goes.