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Oscillations and Waves
Simple Harmonic Motion
When a body or a moving particle repeats its motion along a definite path after regular intervals of time its motion is said to be Periodic Motion and interval of time is called time period (T). The path of periodic motion may be linear, circular, elliptical or any other curve. For example rotation of earth around the sun.
To and fro type of motion is called Oscillatory Motion. A particle has oscillatory motion when it moves about stable equilibrium position. It need not be periodic and need not have fixed extreme positions.
The oscillatory motions in which energy is conserved are also periodic. For example motion of pendulum of a wall clock. The force / torque (directed towards equilibrium point) acting in oscillatory motion is called restoring force/torque Damped Oscillations are those in which energy consumed due to some resistive forces and hence total mechanical energy decreases and after some time oscillation will stop.
TYPES OF SHM:
CHARACTERISTICS OF SHM :
(1) Mean Position : It is the position where net force on the particle is zero.
(2) Extreme Point : Point where speed of the particle is zero.
(3) Displacement : It is defined as the distance of the particle from the mean position at that instant.
(4) Amplitude : It is the maximum value of displacement of the particle from its mean position.
Extreme position – Mean position = Amplitude. It depends upon the energy of the system.
(5) Frequency : The frequency of SHM is equal to the number of complete oscillations per unit time.
(6)Time Period : Smallest time interval after which the oscillatory motion gets repeated is called time period.
Waves is distributed energy or distributed "disturbance (force)"
Following points regarding waves :
(i) The disturbance (force) is transmitted from one point to another.
(ii) The energy is transmitted from one point to another.
(iii) The energy or distrubance passes in the form of wave without any net displacement of medium.
(iv) The oscillatory motion of preceding particle is imparted to the adjacent particle following it.
(v) We need to keep creating disturbance in order to propagate wave (energy or disturbance) continuously.
The waves are classified under two high level headings :
Mechanical waves :
The motion of the particle constituting the medium follows mechanical laws i.e.
Newton's laws of motion. Mechanical waves originate from a disturbance in the medium (such as a stone dropping in a pond) and the disturbance propagates through the medium. The force between the atoms in the medium are responsible for the propagation of mechanical waves. Each atom exerts a force on the atoms near it, and through this force the motion of the atom is transmitted to the others.
The atoms in the medium do not experience any net displacement. Mechanical waves is further classified in two categories such that
1. Transverse waves (waves on a string)
2. Longitudnal waves (sound waves)
Non Mechanical waves :
These are electro magnetic waves. The electromagnetic waves do not require a medium for propagation. Its speed in vacuum is a universal constant. The motion of the electromagnetic waves in a medium depends on the electromagnetic properties of the medium.
If the disturbance travels in the x direction but the particles move in a direction, perpendicular to the x axis as the wave passes it is called a transverse waves.
Longitudinal waves are characterized by the direction of vibration (disturbance) and wave motion. They are along the same direction. It is clear that vibration in the same direction needs to be associated with a "restoring" mechanism in the longitudinal direction.
Describing Waves :
Two kinds of graph may be drawn displacement - distance and displacement-time.
A displacement-distance graph for a transverse mechanical waves shows the displacement y of the vibrating particles of the transmitting medium at different distance x from the source at a certain instant i.e. it is like a photograph showing shape of the wave at that particular instant. The maximum displacement of each particle from its undisturbed position is the amplutude of the wave.
Sound is type of longitudnal wave. In general majority of longitudinal waves are termed as sound waves. Sound is produced by a vibrating source, like when a gong of a bell is struck with a hammer, sound is produced. The vibrations produced by gong are propagated through air, through air these vibrations reach to the ear and ear drum is set into vibrations and these vibrations are communicated to human brain. By touching the gong of bell by hand, we can feel the vibrations.
PROPAGATION OF SOUND WAVES
Sound is a mechanical three dimensional and longitudinal wave that is created by a vibrating source such a guitar string, the human vocal cords, the prongs of a tuning fork or the diaphragm of a loudspeaker. Being a mechanical wavs, sound needs a medium having properties of inetia and elasticity for its propagation. Sound waves propagate in any medium through a series of periodic compressions and rarefactions of presure, which is produced by the vibrating source.
When a longitudinal wave propagated in a gaseous medium, it produces compression and rarefaction in the medium periodically. The region where compression occurs, the pressure is more than the normal pressure of the medium. Thus we can also describe longitudinal waves in a gaseous medium as pressure waves and these are also termed as compression waves in which the pressure at different point of medium also varies periodically with their displacements. Let us discuss the propagation of excess pressure in a medium in longitudinal wave analytically.
CONSTRAINED MOTION :
1.1 String constraint :
When the two object are connected through a string and if the string have the following properties:
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