Orbiting planets and pendulum illustrating classical mechanics principles

Simple harmonic motion

PHYS 201 · Gravity Oscillations and Lagrangian

Simple harmonic motion occurs when acceleration is proportional to displacement and opposite in direction. This lesson covers spring-mass systems, energy, phase, and small oscillations.

Key equations

a=-\omega^2x\frac{d^2x}{dt^2}+\omega^2x=0x(t)=A\cos(\omega t+\phi)v(t)=-A\omega\sin(\omega t+\phi)a(t)=-A\omega^2\cos(\omega t+\phi)F=-kx\omega=\sqrt{\frac{k}{m}}T=2\pi\sqrt{\frac{m}{k}}E=\frac{1}{2}mv^2+\frac{1}{2}kx^2E=\frac{1}{2}kA^2v_{max}=A\omega\sin\theta\approx\thetaT=2\pi\sqrt{\frac{L}{g}}

Learning objectives

Identify the differential equation for simple harmonic motion.
Interpret amplitude, angular frequency, phase, and period.
Analyze spring-mass and small-angle pendulum oscillations.
Describe energy exchange in ideal SHM.

The defining feature

Simple harmonic motion, or SHM, occurs when an object's acceleration is proportional to its displacement from equilibrium and opposite in direction:

$a=-omega^2x$

Since acceleration is the second derivative of position,

$rac{d^2x}{dt^2}=-omega^2x$

$rac{d^2x}{dt^2}+omega^2x=0$

This differential equation is the mathematical signature of SHM.

Sinusoidal solution

The general solution can be written

$x(t)=Acos(omega t+phi)$

Here $A$ is amplitude, $omega$ is angular frequency, and $phi$ is phase constant. The amplitude is the maximum displacement from equilibrium. The phase determines the initial state of the motion.

Velocity and acceleration are

$v(t)=-Aomegasin(omega t+phi)$

$a(t)=-Aomega^2cos(omega t+phi)$

Since $x(t)=Acos(omega t+phi)$ , the acceleration relation $a=-omega^2x$ follows.

Spring-mass oscillator

For a mass on an ideal spring,

$F=-kx$

Newton's second law gives

$m rac{d^2x}{dt^2}=-kx$

$rac{d^2x}{dt^2}+ rac{k}{m}x=0$

Therefore

$omega=sqrt{ rac{k}{m}}$

The period is

$T= rac{2pi}{omega}=2pisqrt{ rac{m}{k}}$

A larger mass oscillates more slowly, while a stiffer spring oscillates more quickly.

Energy in SHM

The total mechanical energy of an ideal spring-mass oscillator is

$E= rac{1}{2}mv^2+ rac{1}{2}kx^2$

At maximum displacement, speed is zero and energy is all spring potential:

$E= rac{1}{2}kA^2$

At equilibrium, spring potential is zero and speed is maximum:

$E= rac{1}{2}mv_{max}^2$

Thus

$v_{max}=Aomega$

Energy continually shifts between kinetic and potential forms while total energy remains constant.

Phase and initial conditions

Initial position and velocity determine $A$ and $phi$ . If the object starts at maximum displacement with zero velocity, cosine with $phi=0$ is convenient. If it starts at equilibrium moving positive, sine may be more convenient.

Different mathematical forms can describe the same physical motion:

$x(t)=Acos(omega t+phi)$

$x(t)=Ccosomega t+Dsinomega t$

Small-angle pendulum

A simple pendulum approximately undergoes SHM for small angles. Its equation is

$rac{d^2 heta}{dt^2}+ rac{g}{L}sin heta=0$

For small angles in radians,

$sin hetaapprox heta$

$rac{d^2 heta}{dt^2}+ rac{g}{L} heta=0$

Thus

$omega=sqrt{ rac{g}{L}}$

and

$T=2pisqrt{ rac{L}{g}}$

The big idea

Simple harmonic motion appears whenever a system has a restoring effect approximately proportional to displacement. Its motion is sinusoidal, its acceleration points toward equilibrium, and its energy alternates between kinetic and potential. SHM is a foundation for waves, sound, circuits, molecular vibrations, and quantum models.

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