Topic 7.3 Notes – Representing and Analyzing SHM

1. What Simple Harmonic Motion Is

In simple harmonic motion, the net force always points toward equilibrium and grows in magnitude as you move farther away.

For a spring,

$$F = -kx$$

The negative sign means the force is opposite the displacement. Since $F = ma$, this gives

$$a \propto -x$$

That proportional relationship is what makes the motion sinusoidal and periodic.

Core ideas and quantities

• Equilibrium position
  Net force = 0. Displacement $x = 0$.
• Displacement (x)
  Position measured from equilibrium. Can be positive or negative.
• Amplitude (A)
  Maximum value of $|x|$. The “edge” of the motion.
• Period (T)
  Time for one full cycle.
• Frequency (f)
  Cycles per second. $f = \frac{1}{T}$
• Angular frequency (ω)
  $$\omega = 2\pi f = \frac{2\pi}{T}$$

Energy constantly shifts between kinetic and potential, but total energy stays constant in ideal SHM.

2. Displacement, Velocity, and Acceleration in SHM

All three are sinusoidal and locked together in a very specific way.

a. Displacement

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

• Use cosine if it starts at maximum displacement.
• Use sine if it starts at equilibrium.
• The choice depends only on initial conditions.

Key features:

• Maximums and minimums: $x = \pm A$
• Zeros: at equilibrium
• One cycle takes time $T$

On a quiz, if they tell you “released from rest at maximum stretch,” that screams cosine.

b. Velocity

Velocity is the derivative of displacement.

If $x = A\cos(\omega t)$, then

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

Important facts:

• $v_{\text{max}} = A\omega$
• Velocity is zero at ±A
• Velocity is maximum at equilibrium
• Shifted by ¼ period from position

Physically, this makes sense. At the turning point, it stops before reversing direction.

c. Acceleration

From the restoring force:

$$a(t) = -\omega^2 x(t)$$

So acceleration is directly proportional to negative displacement.

• $a_{\text{max}} = A\omega^2$
• Acceleration is zero at equilibrium
• Maximum magnitude at $\pm A$
• Always points toward equilibrium
• Shifted ½ period from displacement

Here’s the most testable snapshot idea:

If you can fill in that table from memory, you understand SHM.

3. Amplitude and Period Relationships

One defining feature of SHM is that period does not depend on amplitude.

Bigger amplitude means:

• Larger maximum speed
• Larger maximum acceleration
• More total energy

But the period stays the same.

For common systems:

Mass-spring:

$$T = 2\pi \sqrt{\frac{m}{k}}$$

Simple pendulum (small angles only):

$$T = 2\pi \sqrt{\frac{L}{g}}$$

Notice what’s missing. No amplitude.

If an AP question says the amplitude doubles, your reflex should be: period unchanged.

4. Graphical Representations of SHM

A lot of exam questions give you graphs instead of equations. You should be able to read amplitude, period, velocity, and acceleration directly from them.

Position-time graph

• Amplitude = peak height (±A on the graph)
• Period = time between peaks (for example, from one crest to the next at T and 2T)
• Slope at a point = velocity

Steepest slope happens at equilibrium, where the object is moving fastest.

Velocity-time graph

The middle curve in the figure below shows the velocity-time graph.

Position, velocity, and acceleration graphs for the same oscillation

• Zero when position is ±A
• Maximum at equilibrium
• Area under curve = displacement change

Acceleration-time graph

In the set below, focus on the acceleration graph.

• Zero at equilibrium
• Maximum magnitude at ±A
• Opposite sign of displacement

Connecting them quickly

• Slope of x-t → velocity
• Slope of v-t → acceleration
• Zeros and peaks line up in predictable shifts

On FRQs, they often give one graph and ask you to sketch another. If you know where zeros and maximums must occur, you don’t need full equations to draw it correctly.