Periodic Motion : It is that motion which is identically repeated after a
fixed interval of time. The fixed interval of time after which the motion is repeated is called period of motion.
For example, the revolution of earth around the sun
Oscillatory Motion or Vibratory Motion : It is that motion in which a body moves to and for or back and forth repeatedly about a fixed point (called mean position), in a definite interval of time.
Simple Harmonic Motion : It is a special type of periodic motion, in which a particle moves to and fro repeatedly about a mean (i.e., equilibrium) position and the magnitude of force acting on the particle at any instant is directly proportional to the displacement of the particle from the mean (i.e., equilibrium) position at that instant i.e. F = –k y.
where k is known as force constant. Here, –ve sign shows that the restoring force (F) is always directed towards the mean position.
Displacement in S.H.M : 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. It can be given by the relation
y = a sin t or y = a cos t
The first relation is valid when the time is measured from the mean position and the second relation is valid when the time is measured from the extreme position of the particle executing S.H.M.
Velocity in S.H.M : It is defined as the time rate of change of the displacement of the particle at the given instant. Velocity in S.H.M. is given by
2 2 2dy d
V a sin t a cos t a 1 sin t a 1 y a
dt dt
2 2
a y
Acceleration in S.H.M : It is defined as the time rate of change of the velocity of the particle at the given instant, i.e.,
dv a dt 2 2 d a cos t a sin t y dt
Time period in S.H.M is given by
displacement inertia factor
T 2 , or T 2
acceleration spring factor
Frequency of vibration in S.H.M.,
1 1 acceleration 1 spring factor
, or v
T 2 displacement 2 inertia factor
Total energy in S.H.M
= P.E. K.E. 1m 2y2 1m 2
a2 y2
1m 2a2 a constant.2 2 2
Expression for time period
(i) in case of simple pendulum T 2 g (iv) Oscillations of a loaded spring T 2 m k
where, m is the mass of body attached at the free end of spring and K is the force constant of spring.
Spring constant (K) of a spring : It is defined as the force per unit extension or compression of the spring.
(i) The spring constant of the combination of two springs in series is
1 2 1 2 k k K k k
(ii) The spring constant of the combination of two springs in parallel's K = k1 + k2
Undamped oscillations : When a simple harmonic system oscillates with a constant amplitude (which does not change) with time, its oscillations are called undamped oscillations.
Damped oscillations : When a simple harmonic system oscillates with a decreasing amplitude with time, its oscillations are called damped oscillations.
Free, forced and resonant oscillations
(a) Free oscillations : When a system oscillates with its own natural frequency without the help of an external periodic force, its oscillations are called free oscillations.
(b) Forced oscillations : When a system oscillates with the help of an external periodic force of frequency, other than its own natural frequency, its oscillations are called forced oscillations.
(c) Resonant oscillations : When a body oscillates with its own natural frequency, with the help of an external periodic force whose frequency is the same as that of the natural frequency of the oscillating body, then the oscillations of the body are called resonant oscillations.
A Wave Motion is a form of disturbance which travels through a medium on account of repeated periodic vibrations of the particles of the medium about their mean position, the motion being handed on from one particle to the adjoining particle.
A material medium is a must for propagation of waves. It should possess the properties of inertia, and elasticity. The two types of wave motion are : (i) Transverse wave motion that travels in the form of crests and troughs. (ii) Longitudinal wave motion that travels in the form of compressions
and rarefactions.
Speed of longitudinal waves in a long solid rod is y
where, Y is Young's modulus of the material of solid rod and P is density of the material. The speed of longitudinal waves in a liquid is given by
k
The expression for speed of longitudinal waves in a gas, as suggested by Newton and modified late by Laplace is
p P where C C
P is pressure exerted by the gas.
The speed of transverse waves over a string is given by T m where, T is tension in the string and m is mass of unit length of the string.
Equation of plane progressive waves travelling with a velocity along positive direction of X-axis is
2
y a sin t x
where, is wavelength of the wave, a is amplitude of particle, and x is the distance from the origin.
Superposition principle enables us to find the resultant of any number
of waves meeting at a point. If y , y , y , ...y1 2 3 nare displacements at a point due to n waves, the resultant displacement y at that point is given by y y1 y2 ...yn
On a string, transverse stationary waves are formed due to super- imposition of direct and the reflected transverse waves.
The wavelength of nth mode of vibration of a stretched string is
n
2 L n
and its frequency, vn = n v1
This note is called nth harmonic or (n – 1)th overtone.
Nodes are the points, where amplitude of vibration is zero, In the nth
mode of vibration, there are (n + 1) nodes located at distances (from one end)
L 2L x 0, , ...L
n n
Antinodes are the points, where amplitude of vibration is maximum. In the nth
mode of vibration, there are n antinodes, located at distances (from one end)
2n 1
L 3L 5 L , , , ... L 2 2 n 2n 2n x = n In an organ pipe closed at one end, longitudinal stationary waves are
formed.
fre q u e n cy o f nth mode of vibration, vn = (2n – 1)v1
In an organ pipe open at both ends, antinodes are formed at the two ends, separated by a node in the middle in the first normal mode of vibration and so on. The fundamental frequency in this case is twice the fundamental frequency in a closed organ pipe of same length.
In an open organ pipe, all harmonics are present, whereas in a closed organ pipe, even harmonics are missing.
Beats : Beats is the phenomenon of regular variation in the intensity of sound with time when two sources of nearly equal frequencies are sounded together.
If v1 and v2 are the frequencies of two sources producing beats, then time interval between two successive maxima
1 2
1
v v
time interval between two successive minima
1 2 1 v v Beat frequency = v1 – v2
Doppler's Effect : Whenever there is a relative motion between a source of sound and listener, the apparent frequency of sound heard is different from the actual frequency of sound emitted by the source.
If is actual frequency of sound emitted and v', the apparent frequency, then
L
s v v v ' v v v where, v is velocity of sound in air vs is velocity of source (S) and vL is velocity of listener (L), both moving along SL. Note that velocity along SL is taken positive and velocity along LS is taken negative.
2 1 1 1 2 2 2 2 2 A r F F F A r 2 1 1 1 2 2 2 2 2 A r F F F A r