7.3 Representing and Analyzing SHM

Simple harmonic motion (SHM) is a back-and-forth movement around an equilibrium point. It's characterized by displacement, velocity, and acceleration, which can be described using equations and graphs.

Understanding SHM involves analyzing its key components: amplitude, frequency, and period. These elements help predict an object's position, speed, and direction of motion at any given time during its oscillation.

Displacement, velocity, and acceleration in SHM

When an object undergoes SHM, its displacement, velocity, and acceleration follow predictable patterns that can be described mathematically and graphically.

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Equations for Displacement in SHM

The position of an object in SHM can be described using sine or cosine functions, depending on the initial conditions.

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

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

Understanding the Relationship Between Position, Velocity, and Acceleration

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

• At the equilibrium position, displacement and acceleration are zero, while velocity is at its maximum
• At the maximum displacement (amplitude), velocity is zero and acceleration is at its maximum in the opposite direction of displacement

This relationship creates a continuous cycle where energy transforms between potential and kinetic forms without loss (in an ideal system).

• When the object is at maximum displacement, it has maximum potential energy and zero kinetic energy
• When the object passes through equilibrium, it 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 help identify 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 allows you to determine the stage of the oscillation and the direction of motion. For example, if you see that displacement is zero and velocity is at a positive maximum, you know the object is passing through equilibrium moving in the positive direction.

Amplitude and Period in SHM

The period of oscillation is independent of the amplitude in SHM, which is one of its defining characteristics.

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

• 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 primarily 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 large amount, it will take the same time to complete one full swing (assuming the angle remains small).

Graphical Analysis of SHM

The periodic nature of SHM creates distinctive graphical patterns that help visualize 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

On AP Physics 1, 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.

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

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 — 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.