More Oscillations Physics 202 Professor 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 oscillates As one increases the other decreases Energy must be conserved

SHM Energy Conservation

Potential Energy Potential energy is the integral of force From our expression for x U=½kx m 2 cos 2 (  t+  )

Kinetic Energy Kinetic energy depends on the velocity, K=½mv 2 = ½m  2 x m 2 sin 2 (  t+  ) Since  2 =k/m, K = ½kx m 2 sin 2 (  t+  ) The total energy E=U+K which will give: E= ½kx m 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 xmxm angle =  t+  Particle moving in circle of radius x m viewed edge-on: cos (  t+  )=x/x m x=x m cos (  t+  ) x(t)=x m 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