Oscillations, Waves and Optics
Jan, 2024, ASTU
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ASTU Cover Page - 1/32 Jan – 2024
Simple Harmonic Motion
Periodic or oscillatory motion is motion of an object that regularly returns to a given
position after a fixed time interval.
If something is oscillating (vibrating) this means that it is moving backwards and
forwards, up and down, side to side, and in and out around some central position
(equilibrium point).
Simple harmonic motion (SHM) is a special type of
m
periodic motion or oscillation where;
motion is about an equilibrium position at which
point no net force acts on the system,
the restoring force is directly proportional to the
displacement 𝒙 from the equilibrium position
acts in the direction opposite to that of
displacement.
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Simple Harmonic Motion
Velocity as a Function of Position
Conservation of energy provides a simple method of deriving an expression for the velocity
of an object undergoing periodic motion as a function of position.
This expression shows that the object’s speed is a maximum
at 𝑥 = 0 and is zero at the extreme positions 𝑥 = ±𝐴.
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Simple Harmonic Motion
Period and Frequency
The frequency of the periodic motion of a mass on a spring is
The angular frequency (𝝎) is
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Simple Harmonic Motion
First condition
The solution is
Second condition
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Simple Harmonic Motion
Example 1
‽ A 0.2kg block connected to a light spring for which the force constant is 5 N/m is free to
oscillate on a frictionless, horizontal surface. The block is displaced 5 cm from
equilibrium and released from rest. A) Find the period of its motion. B) Determine the
maximum speed and acceleration of the block. C) Express the position, velocity, and
acceleration as functions of time.
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Simple Harmonic Motion
Example 2
‽ What if the block were released from the same initial position, 𝑥𝑖 = 5 𝑐𝑚, but with an initial
velocity of 𝑣𝑖 = −0.100 𝑚/𝑠?
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Simple Harmonic Motion
Energy of the Simple Harmonic Oscillator
The kinetic energy of the block is
The elastic potential energy stored in the
spring for any elongation 𝑥 is given by
Total mechanical energy
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Simple Harmonic Motion
Example 3
‽ A 0.500kg cart connected to a light spring for which the force constant is 20.0 N/m oscillates on a frictionless,
horizontal air track. A) Calculate the maximum speed of the cart if the amplitude of the motion is 3.00 cm. B)
What is the velocity of the cart when the position is 2.00 cm? C) Compute the kinetic and potential energies of
the system when the position of the cart is 2.00 cm.
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Simple Harmonic Motion
The Simple Pendulum
The forces acting on the bob are the tension and the weight.
T is the force exerted by the string
mg is the gravitational force
The tangential component of the gravitational force is the
restoring force.
Recall that the tangential acceleration is
The equation for 𝜃 is the same form as for the spring,
with solution
This gives another differential equation (t ) max cos(t )
g 2 L
so the period is T = 2
L g
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Simple Harmonic Motion
Damped Oscillations
In many real systems, nonconservative forces such as friction or air resistance also act and
retard the motion of the system. Consequently, the mechanical energy of the system
diminishes in time, and the motion is said to be damped.
where 𝜔0 = 𝑘/𝑚 represents the angular frequency in the absence of a retarding force (the
undamped oscillator) and is called the natural frequency of the system.
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Simple Harmonic Motion
When the magnitude of the retarding force is small such that
𝑏Τ
2𝑚 < 𝜔0 , the system is said to be underdamped.
A: underdamping: there are a few small oscillations before the oscillator
comes to rest.
When 𝑏 reaches a critical value 𝑏𝑐 such that 𝑏𝑐ൗ2𝑚 = 𝜔0 , the
system does not oscillate and is said to be critically damped.
B: critical damping: this is the fastest way to get to equilibrium.
If the medium is so viscous that the retarding force is large
compared with the restoring force that is, if 𝑏Τ2𝑚 > 𝜔0 , the
system is overdamped.
C: overdamping: the system is slowed so much that it takes a long time to
get to equilibrium.
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Simple Harmonic Motion
Forced Oscillations
Forced vibrations occur when there is a periodic driving force. This force may or may
not have the same period as the natural frequency of the system.
If the frequency is the same as the natural frequency, the amplitude becomes quite large.
This is called resonance.
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Wave Motion
A wave is a disturbance that carries energy
from place to place.
A wave does NOT carry matter with it! It just
moves the matter as it goes through it.
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Wave Motion
Some waves do not need matter (called a “medium”) to be able to move (for example,
through space). These are called electromagnetic waves (or EM waves).
Some waves MUST have a medium in order to move. These are called mechanical waves.
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Wave Motion
Type of wave
1. Transverse waves: Waves in which the medium moves at right angles to the direction of the
wave.
2. Compressional (or longitudinal) waves: Waves in which the medium moves back and
forth in the same direction as the wave.
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Wave Motion
Crest: the highest point of the wave. Frequency (f): how many waves go past a point
in one second; unit of measurement is hertz (Hz).
Trough: the lowest point of the Wavelength (λ): The distance between one point
wave. on a wave and the exact same place on the next
wave.
Compression: where the particles Amplitude (A): how far the medium moves from
are close together rest position (where it is when not moving).
Period (T): is the time it takes for one cycle to
Rarefaction: where the particles are complete.
spread apart
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Wave Motion
Traveling wave
The function describing the positions of the elements
of the medium through which the sinusoidal wave is
traveling can be written
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Wave Motion
Reflection and Transmission of Waves
A wave reaching the end of its medium, but where the medium is still free to
move, will be reflected (b), and its reflection will be upright.
A wave hitting an obstacle will be reflected (a), and its reflection will be inverted.
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Wave Motion
In this situation part of the wave is reflected, and
part of the wave is transmitted.
Part of the wave energy is transferred to the more
dense medium, and part is reflected.
The transmitted pulse is upright, while the reflected
pulse is inverted.
The speed and wavelength of the reflected wave
remain the same, but the amplitude decreases.
The speed, wavelength, and amplitude of the
transmitted pulse are all smaller than in the incident
pulse.
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Wave Motion
Reflection;
Bounce off of wave.
Occurs when a waves reaches an obstacle/barrier.
Traveling in the opposite direction.
Its frequency does not change.
Obey first law of reflection of light.
The angle of incident = the angle of reflection
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Wave Motion
Diffraction;
A spreading of wave after passing the edge of an obstacle or gap/slit.
Narrow gap has more effect.
Wide gap has less effect.
Large obstacle, small wavelength =
low diffraction
Small obstacle, large wavelength =
large diffraction
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Wave Motion
Refraction;
breaking of wave.
Is a measure of the extent to which a medium reflects light.
Change speed and direction as they move from one material to another.
The wavelength may decrease/increase.
It use in lenses, camera, telescope.
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Wave Motion
The superposition principle says that when two waves pass through the same point, the
displacement is the arithmetic sum of the individual displacements.
In the figure below, (a) exhibits destructive interference and (b) exhibits constructive
interference.
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Wave Motion
Interference;
Add up or cancel out.
There are two types: constructive and destructive.
Constructive; Destructive;
If two waves are in phase with each other If two waves are out of phase with each
they combine to make a bigger wave. other they cancel each other, so that there is
Constructive interference occurs when the no wave.
crests of one wave are over the crests of
another wave
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Light and Optics
light travels in a straight-line path in a homogeneous medium, until it encounters a boundary
between two different materials.
The reflection of light from such a smooth surface
is called specular reflection.
The reflection from any rough surface is known
as diffuse reflection.
The angle of reflection equals the angle
of incidence:
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Light and Optics
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Light and Optics
The Law of Refraction
The index of refraction, 𝑛, of a medium is defined as:
As light travels from one medium to another,
its frequency doesn’t change.
Snell’s law of refraction.
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Light and Optics
At some particular angle of incidence 𝜃𝑐 , called the critical angle, the refracted light ray
moves parallel to the boundary so that 𝜃2 = 90𝑜 . For angles of incidence greater than 𝜃𝑐 ,
the ray is entirely reflected at the boundary.
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Light and Optics
Example 4
‽ A light ray of wavelength 589 nm (produced by a sodium lamp) traveling through air is
incident on a smooth, flat slab of crown glass at an angle 𝜃1 = 30𝑜 to the normal. (a) Find
the angle of refraction, 𝜃2 . (b) At what angle 𝜃3 does the ray leave the glass as it re-enters
the air?
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Image Formation
Images are formed at the point where rays of light actually intersect or where they appear to
originate.
The image formed by an object placed in front
of a flat mirror is as far behind the mirror as
the object is in front of the mirror.
The lateral magnification M is
defined as:
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End
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