OSCILLATIONS- PART1LS-15/AS 18th,JAN,10 7.32p.m : OSCILLATIONS- PART1LS-15/AS 18th,JAN,10 7.32p.m PERIODIC MOTION SIMPLE HARMONIC MOTION
PERIODIC MOTION : PERIODIC MOTION Periodic motion of a body is that motion which is repeated identically after a fixed interval of time.
The fixed interval of time after which the motion is repeated is call period of motion.
Examples of periodic motion : Examples of periodic motion 1. The revolution of earth around the sun is a periodic motion. Its period of revolution is one year.
Examples of periodic motion : Examples of periodic motion 2. The rotation of earth about its polar axis is a periodic motion. Its period of rotation is one day.
OSCILLATORY MOTION : OSCILLATORY MOTION Oscillatory or Vibratory motion is that motion in which a body moves to and fro or back and forth repeatedly about a fixed point(called mean position or equilibrium position), in a definite interval of time).
Thus a periodic and bounded motion of a body about a fixed point is called an oscillatory or vibratory motion.
OSCILLATION : OSCILLATION The oscillatory motion can be expressed in terms of sine and cosine functions or their combinations. It is due to this reason that the oscillatory motion is called a harmonic motion.
OSCILLATION : OSCILLATION Examples of Oscillatory Motion
1. The motion of the pendulum of a wall clock is oscillatory motion.
2. The motion of the bob of a simple pendulum.
3. The motion of a loaded spring.
4. The motion of liquid contained in U-tube.
DIFFERENCE BETWEEN PERIODIC MOTION AND OSCILLATORY MOTION : DIFFERENCE BETWEEN PERIODIC MOTION AND OSCILLATORY MOTION All oscillatory motions are periodic motions because each oscillatory motion is completed in a definite interval of time. But all periodic motions may not be oscillatory. For example,the revolution of earth around the sun.
HARMONIC OSCILLATIONS : HARMONIC OSCILLATIONS Harmonic oscillation is that oscillation which can be expressed in terms of single harmonic function (i.e. sine function or cosine function).
HARMONIC OSCILLATIONS : HARMONIC OSCILLATIONS In such oscillations, when a body is displaced a little from its equilibrium position and then left to itself, it begins to oscillate to and fro about the mean position under a restoring force.
HARMONIC OSCILLATIONS : HARMONIC OSCILLATIONS Restoring force is always directed towards the mean position and its magnitude at any instant is proportional to the displacement of the body from the mean position at that instant.
HARMONIC OSCILLATIONS : HARMONIC OSCILLATIONS A harmonic oscillation of constant amplitude and of single frequency is called simple harmonic oscillation.
HARMONIC OSCILLATIONS : HARMONIC OSCILLATIONS Mathematically, a simple harmonic
oscillation can be expressed as
HARMONIC OSCILLATION : HARMONIC OSCILLATION y= displacement of body from mean position at any instant t.
a = maximum displacement or amplitude of displacement of the body.
v,T = frequency and time period of harmonic oscillation.
HARMONIC OSCILLATION : HARMONIC OSCILLATION On plotting a graph between y and t as given by(1) get a sine curve as shown in Fig. Similarly on plotting a graph between y and t given by (2), we get a cosine curve.
DEFINITIONS RELATED TO PERIODIC MOTION : DEFINITIONS RELATED TO PERIODIC MOTION TIME PERIOD.It is the least interval of time after which the periodic motion of a body repeats itself.
FREQUENCY.It is defined as the number of periodic motions executed by body per second.
DEFINITIONS RELATED TO PERIODIC MOTION : DEFINITIONS RELATED TO PERIODIC MOTION Angular frequency of a body executing periodic motion is equal to the product of frequency of the body with factor 2?.
DEFINITIONS-PERIODIC MOTION : DEFINITIONS-PERIODIC MOTION DISPLACEMENT
The displacement variable is the deviation of a vibrating or oscillating body from the mean position of the oscillation, with time.
PERIODIC FUNCTIONS : PERIODIC FUNCTIONS Periodic functions are those functions which are used to represent periodic motion. A function f (t) is said to the periodic, if
f (t) = f (t + T)= f (t + 2T)
Since sine and cosine functions are the examples of periodic functions.
PERIODIC FUNCTIONS : PERIODIC FUNCTIONS We know that the value of sine or cosine function repeats after a period of 2? radian.
where ? is called angular frequency
PHASE : PHASE Phase of a vibrating particle at any instant is a physical quantity which completely expresses the position and direction of motion of the particle at that instant with respect to its mean position.
In Oscillatory motion, the phase of a vibrating particle is the argument of sine or cosine function involved to represent the generalised equation of motion of the vibrating particle.
PHASE : PHASE
Initial phase : Initial phase Initial phase or epoch. It is the phase of a vibrating particle corresponding to time t=0.
When t = 0, Its unit is radians.
PHASE DIFFERENCE : PHASE DIFFERENCE PHASE DIFFERENCE between two vibrating particles tells the lack of harmony in the vibrating states of the two particles at a given instant.
It is measured as the difference in phase angles of the two vibrating particles at any instant.
ZERO PHASE DIFFERENCE : ZERO PHASE DIFFERENCE (i) When the two vibrating particles cross their mean positions at the same time, moving in the same direction.
The phase difference between them is zero.
ZERO PHASE DIFFERENCEEXAMPLE : ZERO PHASE DIFFERENCEEXAMPLE
180o -PHASE DIFFERENCE : 180o -PHASE DIFFERENCE (ii) When the two vibrating particles cross their mean position at the same time, moving in the opposite direction and the particle A is ahead of particle B by half vibration.
The phase difference between them is ? rad or 180o
90o. -PHASE DIFFERENCE : 90o. -PHASE DIFFERENCE The phase difference between them is ?/2 rad or 90o.
SIMPLE HARMONIC MOTION : SIMPLE HARMONIC MOTION Simple Harmonic motion is a special type of periodic motion,in which a particle moves to and fro repeatedly about a mean(i.e.equilibrium) position under a restoring force, which is always directed towards the mean (i.e. equilibrium) position and whose magnitude at any instant is directly proportional to the displacement of the particle from the mean(i.e.equilibrium)position at that instant.
SIMPLE HARMONIC MOTIONEXAMPLES : SIMPLE HARMONIC MOTIONEXAMPLES
SIMPLE HARMONIC MOTION : SIMPLE HARMONIC MOTION A particle executing simple harmonic motion along the x-axis between points A and B with O as mean position.
Let at an instant t,the particle be at P, where OP = x, which is the displacement of the particle from the mean position.
The restoring force acting on the particle at that instant is F = - kx
SIMPLE HARMONIC MOTION : SIMPLE HARMONIC MOTION F = - kx
where k is known as force constant.
Its SI unit is Nm-1.
The negative sign shows that the restoring force F is always directed towards the mean position.
The relation above is called force law for SHM.
SIMPLE HARMONIC MOTION : SIMPLE HARMONIC MOTION The displacement of the particle executing simple harmonic motion at an instant can be expressed in terms of one simple harmonic function(i.e.,sine or cosine function).Hence,it is called simple harmonic motion.
SIMPLE HARMONIC MOTION : SIMPLE HARMONIC MOTION If the initial phase of the particle is and at, time t = 0, the particle is at the mean position O, then the displacement y of the particle in SHM at the instant t is given by
SIMPLE HARMONIC MOTION AND UNIFORM CIRCULAR MOTION OR GEOMETRICAL INTERPRETATION OF S.H.M. : SIMPLE HARMONIC MOTION AND UNIFORM CIRCULAR MOTION OR GEOMETRICAL INTERPRETATION OF S.H.M.
GEOMETRICAL DEFINITION OF SIMPLE HARMONIC MOTION. : GEOMETRICAL DEFINITION OF SIMPLE HARMONIC MOTION. Simple Harmonic Motion is defined as the projection of a uniform circular motion on any diameter of a circle of reference.
CHARACTERISTICS OF SIMPLE HARMONIC MOTION : CHARACTERISTICS OF SIMPLE HARMONIC MOTION DISPLACEMENT. The displacement of a particle executing S.H.M. at an instant is defined as the distance of the particle from the mean position at that instant.
Consider a reference particle,moving on a circle of reference of radius a, with uniform angular velocity?.
DISPLACEMENT IN SHM : DISPLACEMENT IN SHM Let the particle start from the point X and trace an angle ? radian in time t as it reaches the point P. Therefore,
Let the projection of the particle P on diameter YOY’ be at M, then M will be executing S.H.M. Here. OM = y ,is the displacement in S.H.M.at time t.
DISPLACEMENT IN SHM : DISPLACEMENT IN SHM
DISPLACEMENT IN SHM : DISPLACEMENT IN SHM If A is the starting position of the particle of reference such that
DISPLACEMENT IN SHM : DISPLACEMENT IN SHM Here - ?o is called the initial phase or epoch of S.H.M.
If B is the starting position of the particle of reference such that ?BOX = ?o and
DISPLACEMENT IN SHM : DISPLACEMENT IN SHM If B is starting position of the particle
Here + ?o is called initial phase or epoch of S.H.M.
DISPLACEMENT IN SHM : DISPLACEMENT IN SHM (ii) Amplitude. The maximum displacement on either side of mean position is called amplitude of motion.
In S.H.M.,the maximum value of sin ? or cos ?=1
Therefore,from equation, the maximum value of displacement (y or x) will be a.
DISPLACEMENT IN SHM : DISPLACEMENT IN SHM Thus the amplitude (i.e. maximum displacement)in simple harmonic motion is equal to the radius of the circle of reference. If S is the span of S.H.M.,then amplitude a = S/2.
Velocity. : Velocity. The velocity of the particle executing S.H.M.at any instant,is defined as the time rate of change of its displacement at that instant.
The velocity in SHM at an instant is the projection of the velocity vector of particle of reference on y-axis or x-axis at that instant.
VELOCITY : VELOCITY At the time t, when the particle of reference is at P, where its projection on y-axis is at M. Here OM = y = displacement in SHM at the instant t. In ?OMP,
VELOCITY : VELOCITY The magnitude of the velocity of the particle of reference at P is v = ?a. It is represented by PA.
The velocity of vibrating point M in SHM at the instant t is the projection of the velocity vector of particle of reference at P on y-axis,where
VELOCITY : VELOCITY
VELOCITY : VELOCITY At mean position, y=0
From V = ? a (Max.)
At the extreme positions, y=a
From V= 0 (Min.) V= 0
VELOCITY : VELOCITY Thus the velocity in S.H.M. is not uniform throughout the motion. It is maximum at the mean position and is minimum at the extreme positions.
The maximum value of velocity is called velocity amplitude in SHM.
The direction of velocity is either towards the mean position or away from the mean position.
ACCELERATION : ACCELERATION (iv) The acceleration of the particle executing S.H.M. at any instant is defined as the time rate of change of its velocity at that instant.
The acceleration in SHM at an instant is the projection of the acceleration vector of the particle of reference on y-axis at that instant.
ACCELERATION : ACCELERATION We know that if a particle of reference is performing a uniform circular motion, it is subjected to radial acceleration directed towards the centre. At the time t, when the reference particle is at P, Fig. where
The magnitude of the acceleration of the particle of reference at P is
ACCELERATION : ACCELERATION It is represented by (PC). The acceleration of the vibrating point M,i.e., at the instant t, is the projection of acceleration vector of reference particle at P on y-axis,i.e., ; Thus, acceleration in SHM is
ACCELERATION : ACCELERATION At mean position, y =0;
From Eq A = 0 (Min.)
At extreme position, y = a;
From Eq A = - ?2a (Max.)
Thus in S.H.M. , the acceleration is also not uniform throughout the motion. It is minimum at the mean position and is maximum at the extreme position.
ACCELERATION : ACCELERATION The maximum value of acceleration is called acceleration amplitude in S.H.M.
In eq negative sign shows that the acceleration (A) is directed opposite to the one in which displacement increases. Since,the displacement increases in the direction away from the mean position(on both the sides) acceleration is always
ACCELERATION : ACCELERATION directed towards the mean position in a S.H.M. Also A ? y because ? is constant. These two characteristics are often used to define S.H.M.
Thus a particle is said to be executing S.H.M.if its acceleration at any instant,is directly proportional to its displacement from the mean position and is always directed towards mean position.
TIME PERIOD : TIME PERIOD It is defined as the time taken by the particle executing SHM to complete one vibration. Neglecting-ve sign,from we have
TIME PERIOD : TIME PERIOD
TIME PERIOD : TIME PERIOD If v is the frequency of vibration of S.H.M.,then
GRAPHICAL REPRESENTATION OF DISPLACEMENT,VELOCITY AND ACCELERATION IN S.H.M. : GRAPHICAL REPRESENTATION OF DISPLACEMENT,VELOCITY AND ACCELERATION IN S.H.M. Consider the displacement of SHM at the instant t, is given by For all your Physics Problems
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GRAPHICAL REPRESENTATION OF DISPLACEMENT,VELOCITY AND ACCELERATION IN S.H.M. : GRAPHICAL REPRESENTATION OF DISPLACEMENT,VELOCITY AND ACCELERATION IN S.H.M. .
GRAPHICAL REPRESENTATION OF DISPLACEMENT,VELOCITY AND ACCELERATION IN S.H.M. : GRAPHICAL REPRESENTATION OF DISPLACEMENT,VELOCITY AND ACCELERATION IN S.H.M. .
OSCILLATION : OSCILLATION (i) All the three quantities displacement velocity and acceleration show harmonic variation with time,having same period.
(ii) The velocity amplitude is ? times the displacement amplitude.
(iii) The acceleration amplitude is ?2 times the displacement amplitude.
OSCILLATION : OSCILLATION (iv) in S.H.M. the velocity is ahead of displacement by a phase angle of ?/2.
(v) In S.H.M. the acceleration is ahead of velocity by a phase angle of ?/2.
(vi) In S.H.M., the acceleration is ahead of displacement by a phase angle of ?.
Slide 65 : For all your Physics Problems
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