7.3 Representing and Analyzing SHM

Simple harmonic motion can be described with the equations x = A cos(2πft) or x = A sin(2πft), and the displacement, velocity, and acceleration all follow sinusoidal patterns that peak at different times. The key skill is reading and connecting x t, v t, and a t graphs to find where an object is in its cycle, and knowing that changing the amplitude does not change the period.

Why This Matters for the AP Physics 1 Exam

This topic builds the representation skills the AP Physics 1 exam rewards. You translate between equations, graphs, and verbal descriptions of the same oscillation, which is exactly the kind of multi-representation thinking that shows up in free-response questions that ask you to create and compare models. Being able to spot maxima, minima, and zeros on a graph and then explain what the object is doing at that instant is a high-value skill for both multiple-choice and written explanations.

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Key Takeaways

• Use x = A cos(2πft) when the object starts at maximum displacement and x = A sin(2πft) when it starts at equilibrium.
• Displacement, velocity, and acceleration peak at different moments: velocity is largest at equilibrium, while acceleration is largest at the turning points.
• Acceleration always points back toward equilibrium, so its sign is opposite the displacement.
• Changing amplitude does not change the period; the period depends on the system (mass and spring constant, or pendulum length).
• Velocity-time graphs are shifted by a quarter period from displacement, and acceleration-time graphs are shifted by half a period.
• Reading zeros and extrema on graphs together tells you position, direction of motion, and whether the object is speeding up or slowing down.

Displacement, Velocity, and Acceleration in SHM

When an object undergoes SHM, its displacement, velocity, and acceleration follow predictable patterns you can describe with equations and graphs.

Equations for Displacement in SHM

The position of an object in SHM can be described using a sine or cosine function, depending on where the object starts.

$x = A \cos(2 \pi ft)$ or $x = A \sin(2 \pi ft)$

• $A$ is the amplitude, the maximum displacement from equilibrium
• $f$ is the frequency, the number of oscillations per second (measured in Hz)
• $t$ is the time elapsed since the motion started

The choice between sine and cosine depends on the object's initial position:

• Use cosine if the object starts at maximum displacement
• Use sine if the object starts at equilibrium

How Position, Velocity, and Acceleration Connect

In SHM, displacement, velocity, and acceleration are linked, but they reach their maximum and minimum values at different times.

• At the equilibrium position, displacement and acceleration are zero, while velocity is at its maximum magnitude.
• At maximum displacement (amplitude), velocity is zero and acceleration is at its maximum magnitude, pointing opposite the displacement.

This creates a continuous cycle where energy moves between potential and kinetic forms without loss in an ideal system.

• At maximum displacement, the system has maximum potential energy and zero kinetic energy.
• Passing through equilibrium, the system has maximum kinetic energy and zero potential energy.

Using Maxima, Minima, and Zeros to Analyze SHM

For SHM, displacement $x$, velocity $v$, and acceleration $a$ each have important maxima, minima, and zeros. These features tell you where the object is in its cycle.

• Displacement is at its maximum at $x = +A$, at its minimum at $x = -A$, and zero at the equilibrium position.
• Velocity is zero at $x = +A$ and $x = -A$ (the turning points), and its maximum magnitude occurs at the equilibrium position. Velocity is positive when the object moves in the $+x$ direction and negative when it moves in the $-x$ direction.
• Acceleration is zero at the equilibrium position and has its maximum magnitude at $x = +A$ and $x = -A$. The sign of acceleration is always opposite the sign of displacement because acceleration always points toward equilibrium.

On graphs, locating the times when $x$, $v$, or $a$ are zero or at maxima/minima lets you determine the stage of the oscillation and the direction of motion. For example, if displacement is zero and velocity is at a positive maximum, the object is passing through equilibrium moving in the positive direction.

Amplitude and Period in SHM

The period of oscillation does not depend on amplitude in SHM, which is one of its defining features.

$T = \frac{1}{f}$

• The amplitude ($A$) can change without affecting the period ($T$).
• For a mass-spring system, the period depends only on the mass and spring constant: $T = 2\pi\sqrt{\frac{m}{k}}$
• For a simple pendulum, the period depends mainly on length (for small angles): $T = 2\pi\sqrt{\frac{L}{g}}$

This independence means that whether you pull a pendulum back a small amount or a larger amount, it takes the same time to complete one full swing, as long as the angle stays small.

Graphical Analysis of SHM

The periodic nature of SHM creates distinctive graphs that help you picture the motion.

Displacement-time graphs show a sinusoidal pattern where:

• The amplitude equals the maximum displacement from equilibrium
• The period equals the time for one complete oscillation
• The frequency equals the number of complete oscillations per second

You should be able to compare displacement-, velocity-, and acceleration-time graphs for SHM and identify how they are shifted relative to one another:

• Velocity-time graphs are sinusoidal but shifted by $\frac{1}{4}$ period relative to displacement. Velocity is zero at maximum displacement and largest in magnitude at equilibrium.
• Acceleration-time graphs are sinusoidal and shifted by $\frac{1}{2}$ period relative to displacement. Acceleration is zero at equilibrium and largest in magnitude at maximum displacement, always pointing toward equilibrium.

Reading these graphs together lets you determine at any moment whether the object is speeding up or slowing down, which direction it is moving, and where it is in its oscillation cycle.

How to Use This on the AP Physics 1 Exam

Free Response

Expect to translate between representations of the same oscillation. A question might ask you to sketch x-t, v-t, and a-t graphs and then explain how they line up. Tie each graph feature to physical meaning: a zero in displacement matches a peak in speed, and a peak in displacement matches a peak in acceleration pointing back toward equilibrium.

Problem Solving

• Pick sine or cosine based on the starting position before plugging in numbers.
• Mark the turning points (x = ±A, v = 0) and the equilibrium crossings (x = 0, v at maximum) first; they anchor your analysis.
• When asked about a specific time, compare it to fractions of the period (T/4, T/2) to quickly find where the object is.

Common Trap

If amplitude changes in a problem, do not change the period. Only the maximum speed, maximum acceleration, and energy change with amplitude.

Practice Problem: Graphical Analysis

Solution:

(a) Since the particle starts at equilibrium (x = 0) at t = 0, we use the sine function: $x = A\sin(2\pi ft) = 0.05 \text{ m} \cdot \sin\left(2\pi \cdot \frac{1}{2} \text{ s}^{-1} \cdot t\right) = 0.05\sin(\pi t)$ meters

(b) Maximum displacement first occurs at t = T/4: $t = \frac{T}{4} = \frac{2 \text{ s}}{4} = 0.5$ seconds

(c) At t = 0.5 seconds (one-quarter of the period):

• Displacement is at its maximum ($x = +A = +0.05$ m). The object is at the positive turning point.
• Velocity is zero because the object is momentarily at rest before reversing direction.
• Acceleration is at its maximum magnitude and directed in the negative direction (toward equilibrium), so acceleration is at its minimum value ($a = -a_{max}$).

At t = 1.0 seconds (one-half of the period):

• Displacement is zero because the object is passing through the equilibrium position.
• Velocity is at its maximum magnitude in the negative direction (minimum value of velocity), meaning the object is moving in the $-x$ direction through equilibrium.
• Acceleration is zero because the object is at equilibrium where the restoring force is zero.

Common Misconceptions

• Larger amplitude means a longer period. Amplitude has no effect on the period in SHM. The period is set by the system, such as mass and spring constant or pendulum length.
• Velocity and acceleration peak at the same time as displacement. They do not. Velocity peaks at equilibrium, where displacement is zero, and acceleration peaks at the turning points, where displacement is largest.
• Acceleration is zero at the turning points. The object is momentarily at rest there (velocity is zero), but acceleration is at its maximum magnitude, pointing back toward equilibrium.
• You can use sine or cosine interchangeably without checking. The correct choice depends on the initial position. Start at equilibrium uses sine; start at maximum displacement uses cosine.
• The object moves fastest at the ends of its motion. It actually moves fastest through equilibrium and slows to a stop at the turning points before reversing.

Related AP Physics 1 Guides

• 7.1 Defining Simple Harmonic Motion (SHM)
• 7.4 Energy of Simple Harmonic Oscillators
• 7.2 Frequency and Period of SHM