Oscillations

Simple Harmonic MotionVelocity and Acceleration in SHM

Force LawTorsion PendulaPhysical Pendula

SHM and Uniform Circular MotionDamped SHM

Forced Oscillations and Resonancepps by C Gliniewicz

The regular back and forth movement of an object is called harmonic motion. Frequency is the number of oscillations that are completed each second. The SI unit of frequency is the hertz (Hz).The period is the time for one cycle to be completed. The period is the inverse of the frequency.Simple Harmonic Motion is a periodic motion in a sinusoidal manner. The position is described by

The amplitude, , is the term in front of the cosine function. It is the maximum distance the object moves from the rest position. The phase angle is the symbol φ. The angular frequency is ω.

The velocity of the particle is simple harmonic motion is the derivative of position function.

max cosx t x t

maxx

22 f

T

max maxcos sind x t d

v t x t x tdt dt

pps by C Gliniewicz

The acceleration is simple harmonic motion is the derivative of the velocity.

In simple harmonic motion, the acceleration is proportional to the displacement, but opposite in sign and the two quantities are related by the square of the angular frequency.

The force causing the motion is described by Hooke’s Law. One can combine the acceleration and Newton’s Second Law to find the force.

Solving for ω,

The potential energy in simple harmonic motion is elastic energy.

The kinetic energy in simple harmonic motion is translational kinetic energy.

2 2max maxsin sin

dv t da t x t x t x t

dt dt

2 2 2F ma m x F kx m x k m 2

k mT

m k

22 2max

1 1cos

2 2U x kx kx t

2 22 2 2 2max max

1 1 1sin sin

2 2 2K t mv m x t kx t

pps by C Gliniewicz

The total energy of the system is the sum of the potential and kinetic energies.

A pendulum can be created by twisting an object.

A real pendulum can have a complicated distribution of mass, much different from a simple pendulum. One needs to know the location of the center of mass, its distance from the pivot and the moment of inertia about the pivot.

Knowing these quantities, one can solve for the acceleration of gravity at any location. Geologists surveying for metals have done these measurements.

2 22 2max max

2 22max

2 2 2max

1 1cos sin

2 21

cos sin2

1cos sin 1

2

E U K kx t kx t

E kx t t

t t E kx

2T

2Tmgh

pps by C Gliniewicz

Simple harmonic motion is just the projection of uniform circular motion on a diameter of the circle in which the motion occurs.

A pendulum swinging through air eventually comes to a stop due to the drag force created by the air. A pendulum in a more viscous fluid would stop even faster. The fluid provides a drag force based on the speed with which the pendulum moves. The drag is a damping force.

For a spring pendulum

2

2

22

max 2

2max

substituting for v and a

0 0 gives a solution

x t cos ' where '4

12

drag

btm

btm

F bv bv kx ma

d x dxma bv kx m b kx

dt dt

k bx e t

m m

E t kx e

pps by C Gliniewicz

A person on a swing being pushed by another is undergoing forced oscillations. Two angular frequencies are involved, the natural frequency of the swing and the driving frequency.

If the two frequencies are equal, then resonance occurs and the swing will reach large amplitude. This resonance can be disastrous in some situations such as Tacoma Narrows Bridge collapse or the collapse of buildings during the Mexico City earthquake.