Kinematics of Simple Harmonic Motion — AP Physics 1
1. What is SHM Kinematics? ★★☆☆☆ ⏱ 3 min
Kinematics of simple harmonic motion (SHM) is the study of how position, velocity, and acceleration of an oscillating object change over time, without analyzing the forces that cause the motion (force analysis is covered in SHM dynamics).
We use standard notation: $A$ for amplitude (maximum displacement from equilibrium), $\omega$ for angular frequency, $f$ for frequency (cycles per second), $T$ for period (seconds per cycle), $\phi$ for phase constant, and $x(t), v(t), a(t)$ for position, velocity, and acceleration at time $t$, respectively. SHM is the foundational prerequisite for all other SHM and wave topics in Unit 7 of AP Physics 1.
2. Defining Condition and Angular Frequency Conversions ★★☆☆☆ ⏱ 3 min
While all periodic motion repeats over a fixed period, only SHM follows the specific proportional relationship between acceleration and displacement that defines it. This relationship separates SHM from other types of periodic motion, and is the starting point for all kinematic analysis.
a(t) = -\omega^2 x(t)
The negative sign is critical: it indicates acceleration always points back toward equilibrium, opposite the direction of displacement from equilibrium. This matches intuition: if you pull a mass on a spring to the right of equilibrium, acceleration pulls left back to center, and vice versa.
Angular frequency is directly related to the more familiar measurable quantities period ($T$, time per cycle) and frequency ($f$, cycles per second) by two core relationships:
\omega = 2\pi f = \frac{2\pi}{T}
Exam tip: Always cancel the negative signs first when solving for $\omega$ from the defining relationship. $\omega$ is always positive, so you will never end up with a negative angular frequency, even if displacement or acceleration are negative.
3. Kinematic Equations for Position, Velocity, and Acceleration ★★★☆☆ ⏱ 4 min
From the defining acceleration-displacement relationship, we can derive equations for position, velocity, and acceleration as functions of time. The general form of the position function (written with cosine for consistency with AP conventions) is:
x(t) = A \cos(\omega t + \phi)
$A$ is the amplitude (maximum displacement from equilibrium, always positive), and $\phi$ is the phase constant, which adjusts the equation to match the initial position and velocity of the oscillator at $t=0$. Velocity is the first derivative of position, acceleration is the second derivative:
v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi)
a(t) = \frac{dv}{dt} = -A \omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)
To find $\phi$, use both initial position and initial velocity at $t=0$. Common cases: if the oscillator is at maximum positive displacement at $t=0$, $\phi = 0$. If the oscillator is at equilibrium moving positive at $t=0$, $\phi = -\pi/2$, simplifying to $x(t) = A \sin(\omega t)$.
Exam tip: Always check both position and velocity at $t=0$ to find the correct sign of the phase constant. Many students stop after only matching the initial position and end up with the wrong phase.
4. Graphical Analysis of SHM Kinematics ★★★☆☆ ⏱ 4 min
AP Physics 1 frequently tests graphical reasoning for SHM, asking you to relate graphs of $x(t)$, $v(t)$, and $a(t)$ or extract properties like amplitude and period from a given graph. The key is understanding the fixed phase differences between the three quantities:
- Acceleration is 180° ($\pi$ radians) out of phase with position: when position is maximum positive, acceleration is maximum negative, and vice versa.
- Velocity is 90° ($\pi/2$ radians) out of phase with position: when position is zero (at equilibrium), velocity is maximum magnitude, and when position is maximum, velocity is zero.
Exam tip: When reading period from a position vs time graph, always measure between two identical points (two consecutive peaks, or two consecutive zero crossings with the same slope direction), not just any two zero crossings.
Common Pitfalls
Why: Students mix up the inverse relationship between period and frequency, confusing which quantity goes in the numerator.
Why: Students remember to take the derivative of cosine, but forget that the derivative of cosine is negative sine.
Why: Students see two zero crossings and assume they are one period apart.
Why: Students confuse SHM kinematics with constant acceleration kinematics, where maximum acceleration is independent of velocity.
Why: Students recall the relationship between tangential speed and angular speed from uniform circular motion, but misapply it to SHM.