More Oscillations

Physics 202Professor Vogel (Professor Carkner’s

notes, ed)Lecture 3

Amplitude, Period and Phase

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

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

Pendulums

A mass suspended from a string and set swinging will oscillate with SHM We will first consider a simple pendulum

where all the mass is concentrated in the mass at the end of the string

Consider a simple pendulum of mass m and length L displaced an angle from the vertical, which moves it a linear distance s from the equilibrium point

The Period of a Pendulum

The the restoring force is:F = -mg sin

For small angles sin We can replace with s/L

F=-(mg/L)s Compare to Hooke’s law F=-

kx k for a pendulum is (mg/L)

Period for SHM is T = 2 (m/k)½

T=2(L/g)½

Pendulum and Gravity

The period of a pendulum depends only on the length and g, not on mass A heavier mass requires more force to

move, but is acted on by a larger gravitational force

A pendulum is a common method of finding the local value of g Friction and air resistance need to be

taken into account

Pendulum Clocks Since a pendulum has a regular period it

can be used to move a clock hand Consider a clock second hand attached to a

gear The gear is attached to weights that try to

turn it The gear is stopped by a toothed mechanism

attached to a pendulum of period = 2 seconds The mechanism disengages when the

pendulum is in the equilibrium position and so allows the second hand to move twice per cycle

Since the period is 2 seconds the second hand advances once per second

Physical Pendulum

Real pendulums do not have all of their mass at one point

Properties of a physical pendulum depend on its moment of inertia (I) and the distance between the pivot point and the center of mass (h), specifically:

T=2(I/mgh)½

Non-Simple Pendulum

Uniform Circular Motion

Simple harmonic motion is uniform circular motion seen edge on

Consider a particle moving in a circle with the origin at the center Viewed edge-on the particle seems to be

moving back and forth between 2 extremes around the origin

The projection of the displacement, velocity and acceleration onto the edge-on circle are described by the SMH equations

Uniform Circular Motion and SHM

x-axis

y-axis

xm angle =t+

Particle movingin circle of radius xm

viewed edge-on:

cos (t+)=x/xm

x=xm cos (t+) x(t)=xm cos (t+)

Particle at time t

Observing the Moons of Jupiter

Galileo was the first person to observe the sky with a telescope in a serious way

He discovered the 4 inner moons of Jupiter Today known as the Galilean moons

He (and we) saw the orbit edge-on

Jupiter and Moons

Apparent Motion of Callisto