The Main Idea 

A spring-mass system oscillates when a mass is displaced from its equilibrium position—the point where the spring’s restoring force balances gravity. This oscillation is called simple harmonic motion.

Setting Up the Differential Equation

Using Newton’s second law and Hooke’s law (restoring force proportional to displacement), the motion satisfies [latex]mx'' + kx = 0[/latex], where [latex]m[/latex] is mass, [latex]k[/latex] is the spring constant, and [latex]x(t)[/latex] is displacement from equilibrium. By convention, positive displacement means the mass is below equilibrium.

Introducing the angular frequency [latex]\omega = \sqrt{k/m}[/latex], we can rewrite this as [latex]x'' + \omega^2 x = 0[/latex]. The general solution is [latex]x(t) = c_1\cos(\omega t) + c_2\sin(\omega t)[/latex].

Key Physical Quantities

The period [latex]T = \frac{2\pi}{\omega}[/latex] is the time for one complete oscillation. The frequency [latex]f = \frac{1}{T} = \frac{\omega}{2\pi}[/latex] counts oscillations per unit time.

Finding the Spring Constant and Mass

The spring constant comes from the equilibrium condition [latex]mg = ks[/latex], where [latex]s[/latex] is the stretch distance. In the English system, [latex]g = 32[/latex] ft/sec²; in metric, [latex]g = 9.8[/latex] m/sec². Use [latex]W = mg[/latex] to find mass from weight.

Alternative Form of the Solution

The solution can be written as [latex]x(t) = A\sin(\omega t + \phi)[/latex], where [latex]A = \sqrt{c_1^2 + c_2^2}[/latex] is the amplitude (maximum displacement) and [latex]\phi[/latex] is the phase shift. To find [latex]\phi[/latex], use [latex]\tan\phi = \frac{c_1}{c_2}[/latex] and verify [latex]\phi[/latex] is in the correct quadrant based on the signs of [latex]c_1[/latex] and [latex]c_2[/latex].

The Main Idea 

In real-world spring-mass systems, friction and air resistance cause oscillations to gradually die off—an effect called damping. The damping force is proportional to velocity and acts opposite to motion, giving us the differential equation [latex]mx'' + bx' + kx = 0[/latex], where [latex]b > 0[/latex] is the damping coefficient.

Three Types of Damping

The characteristic equation [latex]m\lambda^2 + b\lambda + k = 0[/latex] has solutions [latex]\lambda = \frac{-b \pm \sqrt{b^2 - 4mk}}{2m}[/latex]. The value of the discriminant [latex]b^2 - 4mk[/latex] determines the system’s behavior.

• Overdamped ([latex]b^2 > 4mk[/latex]): The system has two distinct negative real roots. The general solution is [latex]x(t) = c_1e^{\lambda_1 t} + c_2e^{\lambda_2 t}[/latex]. The mass returns to equilibrium without oscillating (at most one change of direction). The motion decays quickly.
• Critically Damped ([latex]b^2 = 4mk[/latex]): The system has one repeated negative real root. The general solution is [latex]x(t) = c_1e^{\lambda t} + c_2te^{\lambda t}[/latex]. Like overdamped systems, critically damped systems don’t oscillate. This represents the boundary between oscillatory and non-oscillatory behavior. In practice, it’s nearly impossible to achieve exact critical damping.
• Underdamped ([latex]b^2 < 4mk[/latex]): The system has complex conjugate roots [latex]\alpha \pm \beta i[/latex] where [latex]\alpha < 0[/latex]. The general solution is [latex]x(t) = e^{\alpha t}(c_1\cos(\beta t) + c_2\sin(\beta t))[/latex]. The system oscillates, but the amplitude decreases exponentially over time due to the [latex]e^{\alpha t}[/latex] factor.

Key Observation

For all damped systems, [latex]\lim_{t \to \infty} x(t) = 0[/latex]. The system always returns to equilibrium eventually, regardless of damping type.

The Main Idea 

When an external force acts on a spring-mass system (like bumps in a road or a moving support), we model it with the nonhomogeneous differential equation [latex]mx'' + bx' + kx = f(t)[/latex], where [latex]f(t)[/latex] represents the external force.

Two Parts of the Solution

The general solution has the form [latex]x(t) = c_1x_1(t) + c_2x_2(t) + x_p(t)[/latex], where [latex]c_1x_1(t) + c_2x_2(t)[/latex] is the general solution to the complementary equation and [latex]x_p(t)[/latex] is a particular solution to the nonhomogeneous equation.

Transient vs. Steady-State Behavior

For damped systems, the complementary solution [latex]c_1x_1(t) + c_2x_2(t)[/latex] approaches zero as [latex]t \to \infty[/latex]. This portion is called the transient solution because it only affects the system’s short-term behavior and eventually dies out.

The particular solution [latex]x_p(t)[/latex] is called the steady-state solution because it determines the long-term behavior of the system. After enough time passes, the transient effects disappear and the system’s motion is dominated by the steady-state response to the external force.

Finding the Solutions

Solve the complementary equation [latex]mx'' + bx' + kx = 0[/latex] to get the transient solution. Use the method of undetermined coefficients or variation of parameters to find the particular solution [latex]x_p(t)[/latex]. The steady-state solution typically oscillates at the same frequency as the external forcing function [latex]f(t)[/latex].

The Main Idea 

An RLC series circuit contains a resistor, inductor, and capacitor connected in series. These circuits appear in many applications, including radio tuners. The circuit can be modeled using a second-order differential equation that closely parallels the spring-mass system.

Building the Differential Equation

Using Kirchhoff’s voltage rule (the sum of voltage drops around a closed loop equals zero), we account for voltage drops across each component:

• Inductor: [latex]E_L = L\frac{dI}{dt}[/latex] (proportional to the rate of change of current)
• Resistor: [latex]E_R = RI[/latex] (proportional to current, by Ohm’s law)
• Capacitor: [latex]E_C = \frac{1}{C}q[/latex] (proportional to charge)

Adding these with the voltage source [latex]E(t)[/latex] and using [latex]I = \frac{dq}{dt}[/latex], we get:

[latex]L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = E(t)[/latex]

The Spring-Mass Analogy

This equation has the same mathematical form as the forced spring-mass system [latex]mx'' + bx' + kx = f(t)[/latex]. The correspondences are:

• Inductance [latex]L[/latex] ↔ mass [latex]m[/latex]
• Resistance [latex]R[/latex] ↔ damping coefficient [latex]b[/latex]
• Reciprocal of capacitance [latex]1/C[/latex] ↔ spring constant [latex]k[/latex]
• Voltage source [latex]E(t)[/latex] ↔ external force [latex]f(t)[/latex]

This means all the solution techniques and concepts from spring-mass systems (overdamped, critically damped, underdamped, transient and steady-state solutions) apply directly to RLC circuits. The charge [latex]q(t)[/latex] on the capacitor behaves like the displacement [latex]x(t)[/latex] in a mechanical system.