Waves and Vibrations

Periodic Motion versus Oscillatory Motion

A phenomenon, process or motion, which repeats itself after equal intervals of time, is called

periodic.

If a body moves to and fro repeatedly about a mean position with equal intervals of time then it

is called oscillatory motion. All Oscillatory motions are periodic, but not vice-versa.

Examples of Periodic Motion

Heart beat of a person (Period 0.83 s)

Motion of earth around the sun (1 year)

Motion of Halleys comet around the sun(76 years)

Examples of Oscillatory motion

Oscillation of a simple pendulum

Vibration of a mass attached to a spring

Vibration of a string of a musical instrument

Motion of the earth is periodic but not oscillatory because it is not to and fro.

Oscillatory Motion

For a body to oscillate (or vibrate) three conditions must be satisfied.

(i) The body must have inertia to keep it moving across the

mid-point of its path. which is the equilibrium position of the body

(ii) There must be a restoring force (gravitational, elastic etc) to accelerate the body

towards the mid-point.

(iii) The frictional force acting on the body against its motion must be small.

In general when a body executing oscillations is displaced from its equilibrium position, it

experiences a force directed back toward the equilibrium position. Such equilibrium is called a

stable equilibrium. However, when a displaced body returns to its equilibrium position once

being displaced, it does not become rest there, but overshoots due to the prevailing speed of the

body (due to kinetic energy or inertia), and the result is the back-and-forth motion past the

equilibrium position.

Example 1

Motion of a simple Pendulum

T L

x

increasing

Consider a small mass m oscillating in an arc of a circle. The restoring force ( )towards the

equilibrium position is ,and it is provided by the gravity. If the direction of -increasing

is cosidered as positive, then the restoring force

The tesion T mearly acts to make the mass to move in an arc of a circle. If is small, sin is

very nearly equal to in radians. (Eg. When = 0.1 radians which is nearly about 6o, sin

=0.0998)

Therefore for small angles of oscillations, the restoring force,

But in radians is equal to

(

Or

Since m,g, and L are constants, the restoring force is proportional to the coordinate x for small

displacements,

If

(defines as force constant),

restoring force,

i.e. restoring force is proportional to the displacement.

Example 2

Oscillatory motion of a spiral spring

In general when a body is subjected to a tensile or compressive stress the elongation or the

compression is directly proportional to the magnitude of the applied force, provided the

proportional limit is not exceeded. The proportionality is represented by the force constant k for

each specimen or material, or spring, or similar device.

(Youngs Modulus

Considering the directions of the restoring force (-F), and , the above expression can be

written in the form,

Consider a body of mass m being placed on a smooth horizontal table and attached to a spring of

force constant k. We describe the position of the body with the coordinate x, having origin at o,

and taking x = 0 as the equilibrium position of the body where the spring is neither stretched nor

compressed.

When the body is displaced to the right, x is positive, the spring is stretched, and it exerts a force

(restoring force) on the body toward the left (negative x-direction), toward the equilibrium

position. When it is displaced to the left, x is negative; the spring is compressed, and exerts a

restoring force on the body toward the right (the positive x- direction), again toward the

equilibrium position. Thus the sign of the x-component of the restoring force on the body is

always opposite to that of x itself.

If the spring obeys the Hookes law, then the restoring force on the body is given by,

(1)

where k is the force constant for the spring.

When the body is at a distance x from the equilibrium position, its acceleration a is given by the

Newtons second law.

or

(2)

Now suppose the body is displaced to the right a distance A and released. The spring exerts a

restoring force; the body accelerates with a varying acceleration a in the direction of force, and

moves toward the direction of the equilibrium position with increasing speed. The rate of

increase however, is not constant since the accelerating force becomes smaller and smaller as the

body approaches the equilibrium position at which it becomes zero. As the body possesses its

maximum speed at the equilibrium position, it continues to move toward the left. As soon as the

x

o x

body passes the equilibrium position, the restoring force again begins to act on the body, but now

directed toward right. The speed of the body thus decreases at a rate that increases with the

increasing distance from O. It therefore comes to rest momentarily when its distance from o

becomes A, and repeats its motion.

If there is no loss of energy of the system due to friction, the motion would continue indefinitely.

Motion of a system under the influence of a restoring force which is proportional to the

displacement and directed to a fixed point in its path as described above is called simple

harmonic motion (SHM).

A complete vibration or complete cycle in SHM means one round trip, say from O to A to O to

A and back to O.

The periodic time T or simply period of the motion is the time required for one complete cycle or

vibration.

The frequency f is the number of complete vibrations or cycles per unit time. and the SI

unit of frequency is Hz.

Energy in simple harmonic motion

The elastic restoring force is a conservative force for which the principle of conservation of

energy is applicable. (A force that offers the two way conversion between kinetic and potential

energies is called a conservative force). When the body is at a distance x from o the work done

by this force is stored in the system as potential energy

which is equal to the area

under the F versus x curve.

[Note: When a spring is extended the work done by the restoring force is

, where

is the

average force.

, and it is stored in the spring as potential energy.]

If is the speed of the body when its displacement from o is x, the kinetic energy of the body

F

x 0

Therefore total energy

constant (3)

The total energy E is closely related to the amplitude of the oscillation A mentioned above. In

fact when the speed of the body is zero and the total energy becomes equal to the potential

energy and its value at is

.

and

(4)

This relationship allows us to find the speed of the body at any instant.

Relationship between total energy E, potential energy U and the kinetic energy K for a

body oscillating with SHM

Equations of Simple harmonic motion

The above discussion shows that when a mass m executes a SHM along the x-axis it

moves back and forth around a fixed point in the x direction of its motion. To have a complete

description of the motion we need to know the position, velocity, and acceleration for any given

time. The appropriate relations may be developed with the aid of the circle of reference which is

shown below.

x

U

K

A 0

E

-A

Q

O

A

P

x

The point Q moves counterclockwise around a circle of radius A with a constant angular velocity

(measured in rad. s-1). Point O represents the origin of x-axis. The vector from O to Q

represents the position of point Q relative to point O. The is the angle that the vector makes

with the positive x-axis. This vector, whose horizontal component represents the actual motion

of the point P, is called a phasor. The radius of the circle of reference is always equal to the

amplitude A of the motion.

The point P is called the projection of Q on to the diameter. As the point Q revolves, the point P

moves back and forth along a horizontal line. We shall show that the motion of point P is the

same that of a body under the influence of an elastic restoring force in the absence of friction, i.e.

that it is simple harmonic motion.

The displacement of P at any time t is the distance OP or x; from the figure

If point Q is at the extreme right end of the diameter at time t=0, then = 0 at t=0, and angle at

any time t is given by

Hence (5)

Now , the angular velocity of Q in radians per second, is related to f the number of complete

revolutions of Q per second, by

since there are radians in one complete revolution.

The point P also makes one complete revolution for each revolution of Q. Hence f is also the

frequency of vibration of point P.

The velocity of point P

or (6)

Finally the acceleration of point P is obtained by

i.e. (7)

Now comes the crucial step in showing that the motion of P is simple harmonic. Combining the

expressions for a and x we have, (from eqs (5) & (7)

. (8)

Since is constant, the acceleration a at each instant equals a negative constant times the

displacement x at that instant. But this is just the essential feature of SHM as given in equation

(2) above.

Hence the motion of P is simple harmonic.

Comparing equations (2) and (8), we can make the two situations agreeable precisely if we

choose an angular velocity such that

When that is done the frequency of motion of Q, and thus the actual body, is given by

(9)

Since the period T is the reciprocal of the frequency,

Period T of the SHM is given by

(10)

For example

for a simple pendulum, and it gives

as the period of the simple pendulum.

Equations (5), (6) and (7) are useful to describe the variation of the displacement, velocity and

the acceleration of SHM graphically.

0

2

0 1 0 -1 0 1 0 -1 0 1

Note that the velocity is maximum when the displacement is zero. On the other hand, the

acceleration is zero at the centre and maximum at the ends of the path.

Throughout the above discussion we have assume that the initial position of the body (at time t =

0) is at its maximum positive displacement A, but this is not an essential restriction. One can

assume that the displacement of a particle is zero at time t=0. In such cases the corresponding

equations would take the forms

for displacement,

, for velocity, and

for acceleration,

or one can assume any initial conditions for example, initially(at time t = 0), the the phasor

vector can be at an angle with the positive x-axis, then the angle at time t is given by

Then the equations (5),(6), and (7) above become