OSCILLATION

• Periodic motions, processes, or phenomena are those that repeat themselves at regular
intervals.

Oscillatory Motion:

• If a body moves to and fro around a fixed point at regular intervals of time, it is said to
be oscillatory motion.
• The mean position or equilibrium position is the fixed point around which the body
oscillates.

Simple Harmonic Motion is a type of periodic oscillatory motion in which the particle:

(i) oscillates along a straight line.

(ii) The particle’s acceleration is always directed at a fixed point on the line.

(iii) The particle’s displacement from the oscillations determines the

magnitude of acceleration.

SHM Characteristics

x = A sin (ωt+ Ф) gives the displacement x in SHM at time t, where the three
constants A, and distinguish one SHM from another. A cosine function can
also be used to describe a SHM:

x = A cos (ωt + δ)

• At any point in time, the displacement of an oscillating particle equals the change in its
position vector. “Displacement amplitude” or “simple amplitude” refers to the
maximum value of displacement in an oscillatory motion on either side of its mean
position.

As a result, amplitude A = x max.

• The “time period” of SHM refers to how long it takes an oscillating particle to complete
one full oscillation to and fro around its mean (equilibrium) position.
• Frequency is defined as the number of oscillations per second. It is measured in seconds
per second or Hertz per second. Amplitude has no bearing on frequency or time period.
• The quantity (ωt+ Ф) is known as the SHM phase at time t, and it describes the state of
motion at that time. The quantity is the phase at time f = 0 and is referred to as
the SHM’s phase constant, initial phase, or epoch. In the cosine or sine function, the
phase constant is the time-independent term.
• The energy of a body that performs SHM alternates between kinetic and potential, but
the total energy remains constant. When x is moved away from the equilibrium
position:
• The spring constant of a combination is given by oscillations when two springs with
spring constants k1 and k2 are connected in series.
• Parallel Springs: When one spring is attached to two masses m1 and m2, the spring
constant of the combination is given by k = k1 + k2 • When one spring is attached to two
masses m1 and m2, the spring constant of the combination is given by k = k1 + k2
• Pendulum with a Simple Design: The most common
example of bodies executing S.H.M. is a simple
pendulum. An ideal simple pendulum consists of a
heavy point mass body suspended from a rigid support
by a weightless, extensible, and perfectly flexible
string. It is free to oscillate.
• A simple pendulum’s time period is determined by (i)
the length of the pendulum and (ii) the acceleration
due to gravity (g).
• A second’s pendulum has a time period of two
seconds. The length of a second’s pendulum is found
to be 99.3 cm (= 1 m) when g = 9.8 ms-2.
• If a liquid with density p oscillates in a vertical U-tube
with uniform cross sectional area A, then oscillation is the time period of the oscillation.
• If a cylinder with mass m, length L, material density p, and uniform area of cross section
A oscillates vertically in a liquid with density o, the time period of oscillation is given by
oscillations with Simple Harmonics
• Undamped Simple Harmonic Oscillations: Undamped simple harmonic oscillations
occur when a simple harmonic system oscillates with a constant amplitude that does
not change with time.
• Damped Simple Harmonic Oscillations: Damped simple harmonic oscillations occur
when a simple harmonic system oscillates with a decreasing amplitude over time.
• Forced oscillations occur when a system is compelled to oscillate at a frequency other
than its natural frequency.
• Energy in SHM:-
(a) Kinetic Energy (E k):-
E k = ½ mω 2 (r2 -y 2 ) = ½ mω 2 r2 cos 2 ωt
When, y = 0, then, (E k ) max = ½ mω 2 r2 (maximum)
And
When, y = ±r, then, (E k) min =0 (minimum)
(b) Potential Energy (E p ):-
E p = ½ mω 2 r2 = ½ mω 2 r 2 sin 2 ωt
(E p )max = ½ mω 2 r 2
(c) Total Energy (E):-
 E = E k +E p=½ mω 2 r2 = consereved
E = (E k)max =(E p)max
(i) Displacement in SHM at any instant is given by

y = a sin ωt

or y = a cos ωt

where a = amplitude and

ω = angular frequency.

(ii) Velocity of a particle executing SHM at any instant is given by

v = ω √(a2 – y2)
At mean position y = 0 and v is maximum

vmax = aω
At extreme position y = a and v is zero.

(iii) Acceleration of a particle executing SHM at any instant is given by

A or α = – ω2 y
Negative sign indicates that the direction of acceleration is opposite to the direction in which
displacement increases, i.e., towards mean position.

At mean position y = 0 and acceleration is also zero.

At extreme position y = a and acceleration is maximum

Amax = – aω2
(iv) Time period in SHM is given by

T = 2π √Displacement / Acceleration

Graphical Representation
(i) Displacement – Time Graph (ii) Velocity – Time Graph
(iii) Acceleration – Time Graph

Note The acceleration is maximum at a place where the velocity is minimum and vice – versa.

For a particle executing SlIM. the phase difference between

(i) Instantaneous displacement and instantaneous velocity

= (π / 2) rad

(ii) Instantaneous velocity and instantaneous acceleration

= (π / 2) rad

(iii) Instantaneous acceleration and instantaneous displacement

= π rad

The graph between velocity and displacement for a particle executing SHM is elliptical.

Force in SHM
We know that, the acceleration of body in SHM is α = -ω2 x
Applying the equation of motion F = ma,

We have, F = – mω2 x = -kx

Where, ω = √k / m and k = mω2 is a constant and sometimes it is called the elastic constant.
In SHM, the force is proportional and opposite to the displacement.