Physics C: Mechanics · Unit 6: Oscillations · 14 min read · Updated 2026-05-11
Simple Harmonic Motion — AP Physics C: Mechanics
AP Physics C: Mechanics · Unit 6: Oscillations · 14 min read
1. Definition and Differential Equation of SHM★★☆☆☆⏱ 3 min
Simple harmonic motion (SHM) is periodic motion where the restoring force is directly proportional to displacement from stable equilibrium, and opposite in direction to displacement. SHM is the simplest undamped oscillatory motion, accounting for 12-18% of your total AP Physics C: Mechanics exam score. It appears in both multiple-choice and free-response questions, often paired with forces, energy, or rotational motion.
Applying Newton's second law $F_{\text{net}} = ma = m \frac{d^2x}{dt^2}$ rearranges the force definition to the standard SHM differential equation:
\frac{d^2x}{dt^2} + \omega^2 x = 0
Where $\omega^2 = \frac{k}{m}$ for a mass-spring system. Any motion satisfying this ODE is SHM, regardless of the physical system. The general solution is $x(t) = A \cos(\omega t + \phi)$, where $A$ = amplitude (maximum displacement), $\phi$ = phase constant (adjusts for initial conditions), and $\omega = \frac{2\pi}{T} = 2\pi f$.
Exam tip: If a problem gives you a non-standard system (e.g., mass between two springs, floating object), always derive the ODE from Newton's second law to find $\omega$, do not guess from memorized standard formulas.
2. Kinematics of SHM and Initial Conditions★★★☆☆⏱ 4 min
Once you have the position function for SHM, velocity and acceleration are found via differentiation. For the standard position function $x(t) = A \cos(\omega t + \phi)$:
Velocity (first derivative): $v(t) = -A\omega \sin(\omega t + \phi)$
Key relationships: Maximum speed $v_{\text{max}} = A\omega$, which occurs at equilibrium ($x=0$) when all energy is kinetic. Maximum acceleration $a_{\text{max}} = A\omega^2$, which occurs at maximum displacement ($x = \pm A$) when the restoring force is largest.
To find amplitude $A$ and phase constant $\phi$, use initial conditions ($x_0 = x(0), v_0 = v(0)$). Amplitude follows $A^2 = x_0^2 + \left(\frac{v_0}{\omega}\right)^2$, and phase constant from $\tan\phi = -\frac{v_0}{\omega x_0}$. Always check the quadrant of $\phi$ using the signs of $x_0$ and $v_0$.
Exam tip: Calculators only return arctangent values in the first/fourth quadrants. If $\phi$ is in the second/third quadrant, you must adjust the result manually to get the correct value.
3. Energy in Undamped SHM★★★☆☆⏱ 3 min
For undamped SHM, total mechanical energy is conserved, converting between kinetic energy of the mass and potential energy of the restoring force. For a mass-spring system, $K = \frac{1}{2}mv^2$ and $U = \frac{1}{2}kx^2$.
Substituting SHM position and velocity functions, and using $\omega^2 = \frac{k}{m}$, total energy simplifies to:
E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2
Total energy is constant, proportional to the square of the amplitude. A very useful result from energy conservation is the speed at any displacement: $v = \omega\sqrt{A^2 - x^2}$, which avoids working through phase constants entirely.
Exam tip: If asked for speed at a given displacement, using energy conservation is almost always faster and less error-prone than differentiating position and calculating phase terms.
4. AP-Style Worked Problems★★★★☆⏱ 4 min
Common Pitfalls
Why: Students memorize the sine form for the specific case of starting at equilibrium, and forget to adjust for other starting positions
Why: The SHM result for simple pendulums only applies for small angles where $\sin\theta \approx \theta$, which is required to produce the SHM differential equation
Why: Students focus on the derivative of cosine and forget to differentiate the inner linear term in the argument
Why: Students think gravity changes the restoring force because it shifts the equilibrium position
Why: Students confuse spring combinations with resistor combinations