15-1 Simple Harmonic Motion#

Prompts

What distinguishes simple harmonic motion from other periodic motion? Why is the cosine (or sine) form special?
Explain the physical meaning of amplitude \(x_m\), phase \(\omega t + \phi\), and phase constant \(\phi\). How does \(\phi\) relate to where the particle is at \(t = 0\)?
For SHM, how are velocity and acceleration related to displacement? Where is speed maximum and where is it zero?
A block on a spring oscillates. How do stiffness \(k\) and mass \(m\) affect the period \(T\)? Give physical intuition.
Given \(x(t) = x_m \cos(\omega t + \phi)\), derive \(v(t)\) and \(a(t)\). What is the hallmark relationship between \(a\) and \(x\)?

Lecture Notes#

Overview#

Periodic motion: repeats at regular intervals. Frequency \(f\) = oscillations per second (Hz); period \(T\) = time for one cycle.
Simple harmonic motion (SHM): a special periodic motion whose displacement is a sinusoidal function of time—the simplest oscillatory pattern.
Physical cause: a restoring force proportional to displacement, \(F = -kx\) (Hooke’s law). Spring–block systems are the prototype.
Hallmark: acceleration \(a \propto -x\)—always opposite displacement, magnitude set by \(\omega^2\).

Periodic motion	Simple harmonic motion
Any repeating pattern	Sinusoidal (cos/sin) only
Many possible shapes	\(x = x_m \cos(\omega t + \phi)\)
General	Linear restoring force \(F \propto -x\)

Frequency, period, and angular frequency#

Frequency \(f\): number of complete oscillations per second. SI unit: hertz (Hz) = s\(^{-1}\).
Period \(T\): time for one complete cycle.

(44)#\[ T = \frac{1}{f} \]

Angular frequency \(\omega\): radians per second. One cycle = \(2\pi\) rad, so

(45)#\[ \omega = \frac{2\pi}{T} = 2\pi f \]

Why “angular”?

Oscillation and rotation: Simple harmonic motion is the projection of uniform circular motion onto a diameter.

Imagine a particle \(P'\) moving in a circle of radius \(x_m\) at constant angular speed \(\omega\).
Its projection \(P\) onto the \(x\)-axis executes SHM.
At time \(t\), the angle is \(\omega t + \phi\); the \(x\)-component of the position vector is \(x_m\cos(\omega t + \phi)\)—exactly the displacement formula below.

This explains why SHM is sinusoidal: it is “circular motion viewed edge-on.” The phase \(\omega t + \phi\) is the actual angle of the reference particle on the circle.

Oscillation as the shadow of rotation

Displacement in SHM#

The displacement of a particle in SHM from its equilibrium position is

(46)#\[ x(t) = x_m \cos(\omega t + \phi) \]

The function has three parameters:

Symbol	Name	Meaning
\(x_m\)	Amplitude	Maximum distance from equilibrium; always positive; displacement ranges between \(-x_m\) and \(+x_m\)
\(\omega\)	Angular frequency	Radians per second; sets the rate of oscillation
\(\phi\)	Phase constant	Sets position (and velocity) at \(t = 0\)

The argument \(\omega t + \phi\) is the phase; it advances with time.

Velocity and acceleration#

Differentiating Eq. (46) with respect to time:

(47)#\[ v(t) = \frac{dx}{dt} = -\omega x_m \sin(\omega t + \phi) \]

(48)#\[ a(t) = \frac{dv}{dt} = -\omega^2 x_m \cos(\omega t + \phi) \]

Velocity amplitude \(v_m = \omega x_m\): maximum speed. Acceleration amplitude \(a_m = \omega^2 x_m\): maximum magnitude of acceleration.

Location	Speed	Acceleration magnitude
Extreme points (\(x = \pm x_m\))	Zero	Maximum
Center (\(x = 0\))	Maximum	Zero

Physical picture

At the extremes, the particle has been slowed to a stop so motion can reverse—maximum force, maximum acceleration. At the center, the particle rushes through equilibrium—maximum speed, zero force (spring unstretched).

The hallmark of SHM#

Comparing Eq. (46) and Eq. (48):

(49)#\[ a(t) = -\omega^2 x(t) \]

In SHM, acceleration is always proportional to displacement and opposite in sign. The constant of proportionality is \(\omega^2\).

Parameters vs. initial conditions

Hallmark: If you see \(a \propto -x\) in any oscillating system (mechanical, electrical, tidal), the motion is SHM. The coefficient of \(x\) gives \(\omega^2\).

Among the three parameters:

\(\omega\) is a dynamical property fixed by the system (stiffness and inertia)—you cannot change \(\omega\) without changing the physical setup.
\(x_m\) and \(\phi\) are set by the initial conditions \(x(0)\) and \(v(0)\). Two initial conditions determine two parameters; the degrees of freedom match.

Common initial-condition types:

\(\phi\)	Initial condition	\(x(0)\)	\(v(0)\)
\(0\)	Released from rest at right	\(x_m\)	\(0\)
\(\pi\)	Released from rest at left	\(-x_m\)	\(0\)
\(\pi/2\)	Through center, moving left	\(0\)	\(-\omega x_m\)
\(-\pi/2\)	Through center, moving right	\(0\)	\(\omega x_m\)

Negative \(\phi\) shifts the cosine curve rightward; positive \(\phi\) shifts it leftward.

Poll: Which is SHM?

Which relationship indicates simple harmonic motion?

(A) \(a = 3x^2\)
(B) \(a = 5x\)
(C) \(a = -4x\)
(D) \(a = -2/x\)

Force law and the spring–block oscillator#

Newton’s second law with Eq. (49):

(50)#\[ F = ma = m(-\omega^2 x) = -(m\omega^2)x \]

For a spring, Hooke’s law is

(51)#\[ F = -kx \]

Comparing: \(k = m\omega^2\), so

(52)#\[ \omega = \sqrt{\frac{k}{m}}, \qquad T = 2\pi\sqrt{\frac{m}{k}} \]

Stiffer spring (larger \(k\)) → higher \(\omega\), shorter \(T\)—rapid oscillations.
Larger mass \(m\) → lower \(\omega\), longer \(T\)—sluggish oscillations.

Linear oscillator

A block of mass \(m\) on a frictionless surface, attached to a spring of constant \(k\), is a linear simple harmonic oscillator. “Linear” means \(F \propto x\) (first power), not \(x^2\) or other nonlinear dependence.

Summary#

SHM: sinusoidal displacement \(x(t) = x_m \cos(\omega t + \phi)\); caused by restoring force \(F = -kx\).
Key quantities: amplitude \(x_m\), angular frequency \(\omega = 2\pi f = 2\pi/T\), phase constant \(\phi\).
Velocity \(v = -\omega x_m \sin(\omega t + \phi)\); acceleration \(a = -\omega^2 x_m \cos(\omega t + \phi)\).
Hallmark: \(a = -\omega^2 x\)—acceleration proportional to displacement, opposite sign.
Spring–block: \(\omega = \sqrt{k/m}\), \(T = 2\pi\sqrt{m/k}\).