The document discusses simple harmonic motion (SHM). It defines SHM as motion where the acceleration of a particle is directly proportional to its displacement and always directed towards the mean position. Linear SHM is given as an example where the displacement follows a straight line path, such as the motion of a spring-mounted mass. Terms related to SHM like amplitude, period, and frequency are defined. The differential equation of SHM is derived by applying Newton's second law to a mass attached to a spring.

1. SANT GADGE BABA AMRAVATI UNIVERSITY, AMRAVATI
SHRI SHIVAJI SCIENCE & ARTS COLLEGE, CHIKHLI
Unit – 3
Simple Harmonic Motion
By-
Dr. P. P. Padghan
Assistant Professor
Department of Physics,
Shri Shivaji Science and Arts College Chikhli

2. Distance is a scalar quantity that during its motion corresponds to how much ground
an object has covered.
Displacement is a quantity of a vector that corresponds to how far out of place an
object is, it is the total direction shift of the object.

3. Velocity
The rate of change of displacement is known as velocity
dt
dx
v 
Acceleration
The rate of change of velocity is known as acceleration.
dt
dv
a 
From the definition of velocity, acceleration can be expressed in term of displacement as
2
2
dt
x
d
a
dt
dx
dt
d
a









4. Periodic motion
The motion which repeats itself after equal interval of time is known as periodic motion.
The motion of hands of clock, oscillations of simple pendulum, up and down motion of
needle of sewing machine at constant speed, revolution of moon around the earth etc.
are some examples of periodic motion.
Oscillatory motion
If a particle undergoing periodic motion, covers the same path back and forth over the
same path is called vibrational or oscillatory motion.
Examples of oscillatory motions are oscillations of simple pendulum, motion of the
prongs of tuning forks, vibrations of string of musical instrument etc.
An oscillatory motion is always periodic but periodic may or may not be oscillatory e.g.
the motion of planet around Sun is periodic but not oscillatory

5. Simple harmonic motion
If acceleration of the particle in periodic motion is directly proportional to its displacement and
it is always directed towards mean position, then the motion of that particle is said to be Simple
Harmonic Motion (S.H.M.)
Simple harmonic motion can be broadly classified into two classes namely linear simple
harmonic motion and angular simple harmonic motion.
Linear Simple Harmonic Motion: The motion is said to be linear simple harmonic motion, if the
displacement of a particle executing SHM is linear i.e. displacement is straight line path.
The examples are the motion of simple pendulum, the motion of point mass tied with a spring,
etc.
Angular Simple Harmonic Motion: The motion is said to be angular simple harmonic motion, if
the displacement of a particle executing SHM is angular.
The examples of angular S.H.M. are torsional oscillations and oscillations of compound
pendulum.

6. Terms Related Simple Harmonic Motion (S.H.M.):
1) Amplitude: The magnitude of maximum displacement of a particle from its equilibrium
position or mean position is its amplitude. Its S.I. unit is the metre.
2) Time Period: The time taken by a particle to complete one oscillation is its time period.
Therefore, period of S.H.M. is the least time after which the motion will repeat itself. Thus,
the motion will repeat itself after nT, where n is an integer.
3) Frequency: Frequency of S.H.M. is the number of oscillations that a particle performs per
unit time. S.I. unit of frequency is hertz or r.p.s (rotations per second).
4) Phase: It is the physical quantity that expresses the instantaneous position and direction of
motion of an oscillating system.

7. Q
P

8. Linear Simple Harmonic Motion:
Linear simple harmonic motion is defined as the acceleration of the particle in
periodic motion is directly proportional to its displacement and it is always directed
towards mean position, then the motion of that particle is said to be Simple Harmonic
Motion (S.H.M.).
Consider the body of mass m attached to one end of an ideal spring of force
constant k and free to move on a frictionless horizontal surface as shown in Fig 3.1(a).
When there is no force applied to it, it is at its equilibrium position. Now,
1) If we stretch the body outwards, there is a force exerted by the string on the body that
is directed towards the equilibrium position Fig.3.1(b).
2) If we compress the body inwards, there is a force exerted by the string on the body
towards the equilibrium position Fig.3.1(c).

9. In each case, we can see that the force exerted by the spring is towards the equilibrium
position. This force is called the restoring force.
Let the force be F and the displacement of the body from the equilibrium position be
x then the restoring force acting on body is given by,
where k is known as force constant.
The acceleration ( a ) of the body is given by
where w2 = k/m is constant
Thus in S.H.M., acceleration is proportional to displacement and it is directed towards mean
position
kx
F
x
F




x
a
x
a
m
kx
a
m
F
a
ma
F












2


10. Differential Equation of S.H.M
In linear SHM the force is always directed towards the mean position and its magnitude is
directly proportional to the displacement from the mean position
)
1
(

kx
F
x
F




According to the Newton second law of motion,
0
2
2
2
2




kx
dt
x
d
m
kx
dt
x
d
m
From Eq. (1) and Eq. (2)
)
2
(
2
2

dt
x
d
m
F
ma
F



11. 0
0
0
2
2
2
2
2
2
2






x
dt
x
d
x
m
k
dt
x
d
kx
dt
x
d
m

Dividing by m we get
As we know w2=k/m,
This the differential equation of linear SHM.