Simple Harmonic Motion

Physics 202Professor Vogel (Professor Carkner’s

notes, ed)Lecture 2

Simple Harmonic Motion Any motion that repeats itself in a sinusoidal

fashion e.g. a mass on a spring

A mass that moves between +xm and -xm with period T

Properties vary from a positive maximum to a negative minimum Position (x) Velocity (v) Acceleration (a)

The system undergoing simple harmonic motion (SHM) is a simple harmonic oscillator (SHO)

SHM Snapshots

Key Quantities Frequency (f) -- number of complete

oscillations per unit time Unit=hertz (Hz) = 1 oscillation per second = s-1

Period (T) -- time for one complete oscillation T=1/f

Angular frequency () -- = 2f = 2/T Unit = radians per second (360 degrees = 2

radians) We use angular frequency because the

motion cycles

Equation of Motion

What is the position (x) of the mass at time (t)?

The displacement from the origin of a particle undergoing simple harmonic motion is:

x(t) = xmcos(t + ) Amplitude (xm) -- the maximum displacement

from the center Phase angle () -- offset due to not starting at

x=xm (“start” means t=0) Remember that (t+) is in radians

SHM Formula Reference

SHM in Action Consider SHM with =0:

x = xmcos(t) Remember =2/T

t=0, t=0, cos (0) = 1 x=xm

t=1/2T, t=, cos () = -1 x=-xm

t=T, t=2, cos (2) = 1 x=xm

Phase The phase of SHM is the quantity

in parentheses, i.e. cos(phase) The difference in phase between 2

SHM curves indicates how far out of phase the motion is

The difference/2 is the offset as a fraction of one period Example: SHO’s = & =0 are offset

1/2 period They are phase shifted by 1/2 period

Amplitude, Period and Phase

Velocity If we differentiate the equation for

displacement w.r.t. time, we get velocity: v(t)=-xmsin(t + )

Why is velocity negative?Since the particle moves from +xm to -xm the

velocity must be negative (and then positive in the other direction)

Velocity is proportional to High frequency (many cycles per second)

means larger velocity

Acceleration

If we differentiate the equation for velocity w.r.t. time, we get acceleration

a(t)=-xmcos(t + )This equation is similar to the

equation for displacementMaking a substitution yields:

a(t)=-2x(t)

x, v and a Consider SMH with =0:

x = xmcos(t) v = -xmsin(t) = -vmsin(t)

a = -xmcos(t) = -amcos(t) When displacement is

greatest (cos(t)=1), velocity is zero and acceleration is maximum Mass is momentarily at rest,

but being pulled hard in the other direction

When displacement is zero (cos(t)=0), velocity is maximum and acceleration is zero Mass coasts through the middle

at high speed

Force Remember that: a=-2x But, F=ma so,

F=-m2x Since m and are constant we can write the

expression for force as: F=-kx

Where k=m2 is the spring constant This is Hooke’s Law Simple harmonic motion is motion where force is

proportional to displacement but opposite in sign Why is the sign negative?

Linear Oscillator

A simple 1-dimensional SHM system is called a linear oscillator Example: a mass on a spring

In such a system, k=m2

We can thus find the angular frequency and the period as a function of m and k

mk

ωkm

2πT

Linear Oscillator

Application of the Linear Oscillator: Mass in Free

Fall A normal spring scale does not work in the absence of gravity

However, for a linear oscillator the mass depends only on the period and the spring constant:

T=2(m/k)0.5

m/k=(T/2)2

m=T2k/42

SHM and Energy

A linear oscillator has a total energy E, which is the sum of the potential and kinetic energies (E=U+K)U and K change as the mass oscillatesAs one increases the other decreasesEnergy must be conserved

SHM Energy Conservation

Potential Energy

Potential energy is the integral of force

From our expression for xU=½kxm

2cos2(t+)

2kx21kxdxFdxU

Kinetic Energy

Kinetic energy depends on the velocity,

K=½mv2 = ½m2xm2 sin2(t+)

Since 2=k/m, K = ½kxm

2 sin2(t+)The total energy E=U+K which will

give:E= ½kxm

2

Next Time

Read: 15.4-15.6