# PHYS 1070 Study Guide - Color Vision, Wave, Diffraction Grating

87 views21 pages

Published on 15 Apr 2013

Department

Physics

Course

PHYS 1070

Professor

PHYSICS TEXTBOOK NOTES

STUDY GUIDE 1

Mathematics of travelling and standing waves

Vibrations and Waves

1.2 Simple Harmonic Motion

Many objects in nature are subject when displaced slightly from their equilibrium position, to a force that is proportional to the

displacement from equilibrium. If displaced slightly from equilibrium and then released, they will undergo a back and forth oscillatory

motion known as simple harmonic motion (SHM).

When the mass is at the equilibrium position (x=0), the spring exerts no force on it and if the mass is released from rest at eq, it

remains there. If it is displaced to the right or left (+/- x direction) by a force F applied, the spring is stretched or compressed and

exerts a force F on the mass in the direction opposite to the direction of displacement (ie force is directed back toward x=0). This is

the restoring force. F(x)= -kx. A stiff spring has a large k and a soft spring has a small one. F and x are N and m, k= N/m

(Expressed by a linear graph)

Elastic Potential Energy

Potential energy is expressed in joules. Suppose a force is applied such that the mass is displaced from equilibrium at a constant

velocity. With constant velocity, acceleration of the mass is zero so the net force must be zero (F=ma). Thus, the applied force must be

equal to the force exerted by the spring but in the opposite direction. As the mass moves, work is done by the applied force. If the

force is constant, W= Fapp x delta x. Work is area under a Fapp,x vs x graph. Area= 1/2kx^2.

W= ½ kx^2

This work is positive for elongation of the spring (x>0) and for compression (x<0), and may be considered as elastic potential energy U

stored in the spring. U=1/2kx^2. Elastic potential energy refers to energy that is stored in any object as a result of twisting, stretching

or compressing.

Angular Frequency, Period and Frequency of SHM

If the mass is pulled aside from eq to x=A, in a truly frictionless system, the mass will oscillate indefinitely between the positions x=A

and x=-A.

x=Asin(wt) (in radians)

w is a constant related to the force constant k and the mass m and A is the amplitude of the oscillation which the max distance that

the mass moves away from eq.

w=angular frequency of the oscillation and it is measured in rad/s. Set calculator to radians

W is closely related to the period, T, of the oscillation, which is the time for one complete oscillation of cycle of the particle (from x=A

to x=-A back to x=A). During a time interval of one period, the argument (wt) of the sine function in x=Asin(wt) must increase by 2pi

radians in order that the value of x to return to the value at the beginning of the period. (deltaw=2pi).

W=2pi/T

The number of complete oscillations per second is the frequency. F=1/T (s^-1).

W=2pif

T=2pisqrroot(m/k)

Example

The vibration of the tympanic membrane in the ear is essentially shm. A tympanic membrane having a mass=2.4x10^-5kg is vibrating

with frequency of 550Hz. What is the force constant associated with the membrane?

What are the angular frequency and period?

Solution

F= 1/2pi x sqrroot(k/m)

K=4pi^2f^2m

=4pi^2(550)^2(2.4x10^-5)

=2.9 x10^2 N/m

w=2pif=2pi(550)=3.5x10^3 rad/s

T=1/f=1/550= 1.8x10^-3

Position, Velocity and Acceleration in SHM

X=Asin(2pit/T)

X=Asin(2pift)

t=0 can be chosen to be at any point in the cylclic motion of the particle. The graph will be sine but it can be shifted up to ½ period to

the left or right with the exact shift depending on just where t=0 is chosen to be

x=Asin(wt+phase angle).

S is dimensionless called the phase angle and the exact value depends on the point in the cycle where t is chosen to be zero (depends

on the amount of shift in the sine curve. If set t=0 and then S=0, sine is unshifted.

Another choice is to let t=0 when the particle is at x=A. This gives A=Asin(S) where S=1 , pi/2. When shifted ¼ period left, it turns to

cos.

Objects that oscillate with the same period and pass through x=0 at the same time in the same direction oscillate in phase (phase

angles must be identical). Amplitudes can be different

If one object is at its maximum positive position when the other is at its negative max, their phase angles differ by pi radians or 180

degrees and they are said to oscillate pi radians out of phase. Range from 0 to +/- radians.

As an object oscillates back and forth, its velocity changes with time t.

Vmax= Aw at equilibrium position

The velocity is zero at those times when the object is farthest from eq. that is x=+/-A.

When x is negative, acceleration is positive and vice versa.

amax=Aw^2

Example

The wind is causing a leaf to vibrate, undergoing shm with a period of 0.40s and an amplitude of 1.2 cm. At t=0 the leaf is passing

through its equilibrium position and travelling in the +x-direction.

What is the equation relating x and t?

Sketch a graph

At t=0.72 seconds, how far is the leaf from the eq.

Solution

Since x and t=0 and the velocity is positive, then x is related to t by a sine function with no phase shift.

x=Asin(wt)

Since period T is given, use w=2pi/T

x=Asin(2pit/T)

x=(0.012)sin(5.0pit)

see 1-11 for graph

x=(0.012)sin((5.0)t)

=(0.012)sin(5 x 0.72)

=-0.011 m

Example

A rubber raft is bobbing up and down beside a dock (SHM). The simple harmonic motion is described by x=0.30sin(5.0t +0.40). What is

the amplitude, period and frequency?

Solution

x=Asin(wt+S)

x=0.30sin(5.0t +0.40)

w=2pif

so f=w/2pi=5.0/2pi=0.80Hz

T=1/f=1/0.80=1.3 seconds

Damped and Forced Harmonic Motion and Resonance

Friction causes the amplitude to decrease with time. This damped harmonic motion. If the friction is large, the particle will not

oscillate at all but will simply return slowly to eq (looks like ski slope). It is possible to attempt to drive a system into oscillation at any

frequency by applying a driving force that varies periodically with time (forced harmonic motion)

If the system is driven at its free oscillation frequency, oscillations of very large amplitude can easily be built up. This is called

resonance. And the free oscillation frequency is resonant frequency. Think of a child’s sing. It has a natural oscillation and it is hard to

push it back and forth at any other frequency. However, gentle pushes applied at the natural oscillating frequency can easily cause the

swing to swing back and forth with a ver large amplitude.

1.3 Traveling Waves

waves on the water. In travelling waves, the disturbance moves forward across the water although the particles move approx. up and

down at one location. The water does not move forward though the wave disturbance does.

Traveling wave-a disturbance which moves through space carrying energy without the bulk forward movement of matter. Sound is

another example. In water, the particles move up and down perpendicular to the direction (horizontal) that the wave travels

(transverse waves). Sound waves, the particles oscillate back and forth in the same direction of the wave (longitudinal waves). Light

is transverse.

Equation of a Traveling Wave

Y is the size of the wave disturbance (displacement). In a water wave, y represent vertical displacement. On these graphs there can

be numerous sine waves for different times (time is a phase shift, graph is a snapshot at a time). The wave is sine and repeats itself

after a length (wavelength). The wavelength is the distance between the two successive crests or troughs. As the wave moves along,

the disturbance y at any particular point at a fixed value of position x oscillates in shm about its eq. Frequency is the number of

wavelengths that pass per second. T=wavelength=1/f

Y=Asin ((2pi/T)t – (2pi/wavelength)x)) for a wave traveling in the POSITIVE DIRECTION

K is the wave number k=2pi/wavelength (m^-1).

Y=Asin(wt +/- kx)

Wave Speed

The speed of a wave can be expressed in terms of wavelength and period. (v=wavelength/T=wavelength x frequency).

The speed depends on the properties of the medium through which it moves. Once a wave is launched, its frequency remains

constant but as it passes through mediums, the speed and thus the wavelength change.

Example

A wave moves along a string in the x direction with speed of 8.0m/s, a frequency of 4.0 Hz and amplitude of 0.050m.

a. wavelength, wave number, period, angular frequency and equation?

b. Value of y for point at x=3/4 m at time 1/8 s

Solution

V= wavelength/f = (8.0)/(4.0)= 2.0m

K= 2pi/wavelength = (2pi)/(2.0)= pi

T=1/f =1/(4.0)= 0.25s

W=2pif= 8pi rad/s

Y=0.050sin(8pit-pix)

Y=(0.050)sin((8pi)(1/8)-(pi)(3/4))

=0.035m

A standing wave is produced when two waves of the same wavelength travel in opposite directions through the same medium. Very

often, one wave is simply the reflection of the other wave from a surface. The two waves interfere with each other, and by the

principle of wave superposition, the resultant wave displacement y is the sum of the displacements of the two interfering waves.

Thus, when a crest from one wave is at the same position as the crest from the other, the waves are undergoing constructive

interference making a supercrest.

Where a crest interferes with a trough, the two waves undergo destructive interference and the resultant is a small displacement. If

the crest and trough are the same size, completely cancelled.

Equation for a Standing Wave

## Document Summary

Many objects in nature are subject when displaced slightly from their equilibrium position, to a force that is proportional to the displacement from equilibrium. If displaced slightly from equilibrium and then released, they will undergo a back and forth oscillatory motion known as simple harmonic motion (shm). When the mass is at the equilibrium position (x=0), the spring exerts no force on it and if the mass is released from rest at eq, it remains there. A stiff spring has a large k and a soft spring has a small one. F and x are n and m, k= n/m (expressed by a linear graph) Suppose a force is applied such that the mass is displaced from equilibrium at a constant velocity. With constant velocity, acceleration of the mass is zero so the net force must be zero (f=ma). Thus, the applied force must be equal to the force exerted by the spring but in the opposite direction.