AP Physics C Mechanics: Oscillations with Differential Equations

Oscillatory motion is the heartbeat of the physical world, from a child on a swing to the vibrations of an atom. For AP Physics C, moving beyond the standard formulas to solve the underlying differential equations transforms your understanding from memorization to mastery. This approach not only cements your knowledge of simple harmonic motion (SHM) but provides the essential mathematical toolkit for analyzing the real-world complexities of damped and driven systems, a cornerstone for any future engineering or physics coursework.

Deriving the General Solution from the Differential Equation

The defining characteristic of simple harmonic motion is a restoring force directly proportional to the displacement but in the opposite direction. For a mass-spring system, this is Hooke's Law: $F=-kx$. Applying Newton's Second Law, $F=m\frac{d^{2}x}{d{t}^2}$, we arrive at the equation of motion: $$m\frac{d^{2}x}{d{t}^2}=-kx.$$ Dividing both sides by the mass $m$, we obtain the canonical form of the SHM differential equation: $$\frac{d^{2}x}{d{t}^2}=-\frac{k}{m}x.$$ We define the angular frequency $\omega$ as $\omega =\sqrt{\frac{k}{m}}$, yielding the clean, foundational equation you must know: $$\frac{d^{2}x}{d{t}^2}=-{\omega }^{2}x.$$

This equation states that the acceleration is proportional to the negative of the displacement. Our task is to find a function $x(t)$ whose second derivative is a negative constant times itself. The sine and cosine functions possess this exact property. The general solution is a linear combination of both: $$x(t)=B\cos (\omega t)+C\sin (\omega t),$$ where $B$ and $C$ are constants determined by initial conditions. Using a trigonometric identity, this sum of sine and cosine can be rewritten as a single cosine function with a phase shift: $$x(t)=A\cos (\omega t+\varphi ).$$ Here, $A$ represents the amplitude, or maximum displacement, and $\varphi$ is the phase constant (or phase angle), which determines the initial position of the oscillation cycle. You can verify this is a solution by taking the second derivative: $\frac{d^2}{d{t}^2}[A\cos (\omega t+\varphi )]=-{\omega }^{2}A\cos (\omega t+\varphi )=-{\omega }^{2}x$.

Applying Initial Conditions to Determine $A$ and $\varphi$

The general solution $x=A\cos (\omega t+\varphi )$ has two unknown constants: the amplitude $A$ and phase constant $\varphi$. A specific physical situation requires two initial conditions, typically the position $x(0)$ and velocity $v(0)$ at time $t=0$. The velocity function is the derivative: $v(t)=\frac{dx}{dt}=-\omega A\sin (\omega t+\varphi )$.

Let's apply the initial conditions:

1. At $t=0$, position: $x(0)=A\cos (\varphi )$.
2. At $t=0$, velocity: $v(0)=-\omega A\sin (\varphi )$.

Given numerical values for $x(0)$ and $v(0)$, you solve for $A$ and $\varphi$. A reliable method is to divide the velocity equation by the position equation: $$\frac{v(0)}{x(0)}=\frac{-\omega A\sin (\varphi )}{A\cos (\varphi )}=-\omega \tan (\varphi ).$$ This lets you solve for $\varphi$: $\tan (\varphi )=-\frac{v(0)}{\omega x(0)}$. You must then use the signs of $x(0)$ and $v(0)$ to choose the correct quadrant for $\varphi$. To find $A$, square both initial equations and add them, utilizing the identity ${\cos }^2\varphi +{\sin }^2\varphi =1$: $$[x(0){]}^2+{\left[\frac{v(0)}{\omega }\right]}^2=A^2{\cos }^2\varphi +A^2{\sin }^2\varphi =A^2.$$ Thus, $A=\sqrt{[x(0){]}^2+{\left[\frac{v(0)}{\omega }\right]}^2}$. This elegant result shows that the amplitude depends on both the initial displacement and the initial speed.

Extending to Damped Oscillations

Real systems lose energy due to friction or drag. We model this with a damping force often proportional to velocity, $F_d=-bv$, where $b$ is the damping coefficient. The new equation of motion becomes: $$m\frac{d^{2}x}{d{t}^2}+b\frac{dx}{dt}+kx=0.$$ Dividing by $m$, we get the standard form: $$\frac{d^{2}x}{d{t}^2}+2\beta \frac{dx}{dt}+{\omega }_0^{2}x=0.$$ Here, ${\omega }_0=\sqrt{k/m}$ is the natural angular frequency (without damping), and $\beta =b/(2m)$ is the damping constant.

The solution to this second-order linear differential equation depends on the strength of damping, defined by the relationship between $\beta$ and ${\omega }_0$:

• Underdamped ($\beta <{\omega }_0$): The system oscillates with exponentially decaying amplitude. The solution is $x(t)=A{e}^{-\beta t}\cos ({\omega }_{d}t+\varphi )$, where the damped frequency is ${\omega }_d=\sqrt{{\omega }_0^2-{\beta }^2}$.
• Critically Damped ($\beta ={\omega }_0$): The system returns to equilibrium in the shortest possible time without oscillating. The solution takes the form $x(t)=(C_1+C_{2}t)e^{-\beta t}$.
• Overdamped ($\beta >{\omega }_0$): The system slowly returns to equilibrium without oscillation. The solution is a sum of two decaying exponentials: $x(t)=e^{-\beta t}[D_1{e}^{\sqrt{{\beta }^2-{\omega }_0^2}t}+D_2{e}^{-\sqrt{{\beta }^2-{\omega }_0^2}t}]$.

Analyzing Driven Oscillations and Resonance

To sustain oscillation against damping, we apply an external driving force, often sinusoidal: $F(t)=F_0\cos (\omega t)$. The equation of motion for a driven, damped oscillator is: $$m\frac{d^{2}x}{d{t}^2}+b\frac{dx}{dt}+kx=F_0\cos (\omega t).$$ The general solution is the sum of the transient solution (the solution to the homogeneous, undriven equation) and the steady-state solution. The transient part decays exponentially over time due to damping, leaving only the steady-state response: $x_s(t)=A\cos (\omega t-\delta )$.

The key result is that the amplitude $A$ and phase lag $\delta$ of the system's response depend on the driving frequency $\omega$: $$A=\frac{F_0/m}{\sqrt{({\omega }_0^2-{\omega }^2{)}^2+(2\beta \omega {)}^2}},\tan \delta =\frac{2\beta \omega }{{\omega }_0^2-{\omega }^2}.$$ Resonance occurs when the driving frequency $\omega$ matches the system's natural damped frequency, leading to maximum amplitude. For light damping, this resonant frequency is approximately ${\omega }_0$. The amplitude at resonance is $A_{res}=\frac{F_0}{2m\beta {\omega }_0}$, which can be very large if damping ($\beta$) is small. This explains phenomena from shattered wine glasses to tuned radio circuits.

Common Pitfalls

1. Misinterpreting the Phase Constant ($\varphi$): A common error is calculating $\varphi =\arctan (-v(0)/(\omega x(0)))$ and taking the calculator's principal value without considering quadrant. The signs of $x(0)$ and $v(0)$ are crucial. For example, if $x(0)$ is positive and $v(0)$ is negative, $\varphi$ is in the first quadrant (a small positive angle), meaning the oscillator starts near +A but moving toward equilibrium.

1. Confusing Angular Frequency ($\omega$), Frequency ($f$), and Period ($T$): Remember the relationships: $\omega =2\pi f=2\pi /T$. In the damped oscillator equation, carefully distinguish the natural frequency ${\omega }_0$, the damping constant $\beta$, and the damped frequency ${\omega }_d$. Using ${\omega }_0$ in the damped solution formula will lead to an incorrect result.

1. Forgetting the Transient Solution in Driven Systems: When solving the driven oscillator equation, the complete solution includes both transient and steady-state parts. For many physics C problems, they ask for the long-term "steady-state" behavior, but it's vital to understand that the initial motion is a superposition of both, especially when applying initial conditions to the full general solution.

1. Incorrectly Applying the Resonance Formula: The maximum amplitude occurs not exactly at $\omega ={\omega }_0$ for a damped driven oscillator, but at ${\omega }_{max}=\sqrt{{\omega }_0^2-2{\beta }^2}$. However, for light damping ($\beta <<{\omega }_0$), ${\omega }_{max}\approx {\omega }_0$. Know which formula to use based on the problem's context.

Summary

• The fundamental differential equation for simple harmonic motion is $\frac{d^{2}x}{d{t}^2}=-{\omega }^{2}x$, and its general solution is $x(t)=A\cos (\omega t+\varphi )$.
• The amplitude $A$ and phase constant $\varphi$ are determined by initial conditions: $A=\sqrt{[x(0){]}^2+[v(0)/\omega {]}^2}$ and $\tan \varphi =-v(0)/(\omega x(0))$, with quadrant chosen using signs of $x(0)$ and $v(0)$.
• Damped oscillations are governed by $\frac{d^{2}x}{d{t}^2}+2\beta \frac{dx}{dt}+{\omega }_0^{2}x=0$, leading to underdamped ($\beta <{\omega }_0$), critically damped ($\beta ={\omega }_0$), or overdamped ($\beta >{\omega }_0$) solutions.
• Driven oscillations with damping follow $\frac{d^{2}x}{d{t}^2}+2\beta \frac{dx}{dt}+{\omega }_0^{2}x=(F_0/m)\cos (\omega t)$, resulting in a steady-state solution $x_s(t)=A\cos (\omega t-\delta )$ where the amplitude $A$ is maximized at resonance.
• Mastering this differential equations framework empowers you to analyze virtually any oscillatory system, from ideal springs to complex electrical circuits.