Topic 7.3 Notes – Representing and Analyzing SHM

1. What Simple Harmonic Motion Is

An object is in simple harmonic motion when its acceleration is proportional to its displacement from equilibrium and always points toward equilibrium.

Mathematically, that statement becomes:

$$\frac{d^2x}{dt^{2}} = -\omega^{2} x$$

That negative sign is everything. If $x$ is positive, acceleration is negative. If $x$ is negative, acceleration is positive. The system is always pulled back.

You’re not expected to prove the solution to this differential equation. You are expected to recognize that its solutions are sinusoidal:

$$x(t) = A\cos(\omega t + \phi) \quad \text{or} \quad x(t) = A\sin(\omega t + \phi)$$

Key quantities

• $A$ → amplitude (maximum displacement)
• $\omega$ → angular frequency (rad/s)
• $f = \frac{\omega}{2\pi}$
• $T = \frac{2\pi}{\omega}$
• $\phi$ → phase constant (sets initial conditions)

The natural frequency is the frequency a system oscillates at when displaced and released.

Examples:

• Mass-spring: $\omega = \sqrt{\frac{k}{m}}$
• Simple pendulum (small angles only): $\omega = \sqrt{\frac{g}{L}}$

If you ever see $a = -\text{constant} \cdot x$, you should immediately think SHM.

2. Displacement, Velocity, and Acceleration in SHM

Take the general position equation:

$$x(t) = A\cos(\omega t + \phi)$$

Everything else comes from derivatives.

Velocity

$$v(t) = -A\omega \sin(\omega t + \phi)$$

Maximum velocity:

$$v_{\max} = A\omega$$

This occurs at equilibrium where $x = 0$. The object is moving fastest as it passes the middle.

Acceleration

$$a(t) = -A\omega^{2} \cos(\omega t + \phi)$$

Since $x = A\cos(\omega t + \phi)$,

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

This equation defines SHM.

Maximum acceleration:

$$a_{\max} = A\omega^{2}$$

This occurs at the turning points $x = \pm A$.

Zeros and Extrema

You should be able to recognize these instantly:

This shows up constantly in conceptual multiple choice and in FRQs where they ask what happens “at the instant the mass passes equilibrium.”

3. Phase and Graphical Representations

The motion is sinusoidal, but the three graphs are not identical. Compare how displacement, velocity, and acceleration line up in time below.

Displacement, velocity, and acceleration vs. time in SHM

Phase relationships

• Velocity is $90^\circ$ ($\pi/2$) out of phase with displacement.
• Acceleration is $180^\circ$ ($\pi$) out of phase with displacement.
• When $x$ is max → $v = 0$
• When $x = 0$ → $|v|$ is max
• When $x > 0$ → $a < 0$

If you’re given only a graph, you should be able to determine:

• Amplitude from vertical height
• Period from one full cycle
• Frequency from $1/T$
• Angular frequency from $2\pi/T$

Also, the slope of an $x$ vs. $t$ graph gives velocity. Positive slope means moving in the + direction.

4. Amplitude and Period

One of the most tested conceptual ideas:

Changing amplitude does not change period in ideal SHM.

For example: $$T = 2\pi\sqrt{\frac{m}{k}} \quad \text{and} \quad T = 2\pi\sqrt{\frac{L}{g}}$$

Amplitude is not in either formula.

If amplitude increases:

• $v_{\max}$ increases because $v_{\max} = A\omega$
• $a_{\max}$ increases because $a_{\max} = A\omega^{2}$
• Period stays the same

This is often hidden in lab-style questions where they change how far something is pulled back.

5. Resonance and Driven Oscillations

If a sinusoidal external force drives the system, the behavior depends on frequency.

Resonance occurs when the driving frequency equals the natural frequency.

At resonance:

• Energy transfer is maximized.
• Amplitude increases dramatically.
• Even a small force can produce large oscillations.

The system “builds up” motion because each push reinforces the motion at the perfect time.

If the driving frequency is far from natural frequency, oscillations stay small.

On free-response, you’re often asked to explain why amplitude increases. The correct reasoning always involves matching the driving frequency to the natural frequency and maximizing energy transfer.