16.105 Physics I : Mechanics

Motion of particles described by vectors

r(x,y,z,t) = x(t) î + y(t) + z(t) k

Motion is the sum of motion along three perpendicular directions

Point-like particle moves according to Newton’s Laws

F = mdp/dt =m a

16.107 Physics II : Waves and Modern Physics

Oscillations: building blocks are periodic functions

(x,t) = n an cos( knx) cos(nt)

Motion is the sum of periodic functions.

(x,t) is a solution of Schrödinger equation

-(2/2m)2(x,t)/2 x + V(x) (x,t)= i (x,t)/ t

Solutions are wave-like

Chapter 17 Oscillations

• Oscillations are everywhere

• vibrations in your eardrum, cell phone, CD player, quartz watches, heart beat

• some of these vibrations are obvious to us and some are not even detectable

• however, the mathematical description is basically the same

Oscillations

• Some oscillations occur in a medium such as sound waves or water waves

• light, radio waves, x-rays are also oscillatory phenomena but do not involve motion of a medium but rather electric and magnetic fields

• sound waves need a medium

• light waves do not need a medium

Oscillations• Vibrations do not continue

forever unless we continually pump the system

• PendulumclockspringQuartz watch battery

• we need to balance losses due to damping or friction

• energy is converted into heat and lost eg. slinky

• ‘Snapshots’ of oscillating system

• frequency f = # of oscillations each second

• 1 hertz = 1 Hz = 1 s-1

• period T = 1/f is the time for one complete oscillation

Mass

Simple Harmonic Motion

• xm is the amplitude (maximum value)

is the phase angle (determines x(0))“phi” is ? “omega”

• x(t + T) = x(t) ------> (t+T)+ = t + (t+T) = t + 2 ----> T = 2 = 2/T = 2 f angular frequency rads/sec

• recall circular motion =/t

x(t) = xm cos(t + )

Periodic motionor

Harmonic motion

• Motion is repeated at regular intervals

• x(t) = xm cos(t + ) simple harmonic motion

• xm (positive constant) is the amplitude

t + is the phase of the motion is the phase constant(or phase angle)

• x(0) = xm cos( )

Both curves have the same period and phase constant

( ) cos( )mx t x t 2 /T

Both curves have the same amplitude and phase constant

2 /T 2

t + is the phase x(t)= xm cos( t + )points with the same phase have the same value of xa decrease in shifts the red curve to the right

Both curves have the same amplitude and period

xm occurs when phase t+=0

SHM

• xm is a constant

is a constant “phi” is a constant “omega”

t + varies with time t• x(t) varies with time t

• let’s try some examples using maple

x(t) = xm cos(t + )

shm.mws

Velocity

• v(t) = - xm sin(t + )

• But cos(A+B)= cos(A)cos(B)-sin(A)sin(B)

• Hence cos(t + + /2) = - sin(t + )

• v(t) = xm cos(t + + /2)

• phase constant increased --> shift to left

x(t) = xm cos(t + )

Acceleration

• v(t) = xm cos(t + + /2)

• Calculate acceleration

• a(t)= - 2 xm cos(t + )

• a(t)= 2 xm cos(t + + /2 + /2)

• phase constant increased again --> shift left

x(t) = xm cos(t + ) v(t) = - xm sin(t + )