Simple harmonic motion

1. SIMPLE HARMONIC MOTION
Dr. Popat S. Tambade
Associate Professor
Prof. Ramkrishna More Arts, Commerce and Science College
Akurdi, Pune 411 044

2. Oscillations
P. S. Tambade

Content
1.Equilibrium
2.Stable equilibrium
3.Unstable Equilibrium
4.Oscillatory Motion
5.Spring –Mass system
6.Simple harmonic Motion
7.Displacement and velocity
8.Periodic Time
9.Frequency
10.Displacement and Acceleration
11.Energy of SHM
12.Lissajous Figures
13.Angular SHM
14.Simple Pendulum

3. Oscillations
P. S. Tambade

Equilibrium
•Types of equilibriums
1.Stable Equilibrium
2.Unstable equilibrium
3.Neutral equilibrium
The body is said to be in equilibrium at a point
when net force acting on the body at that point is
zero.
C

4. Oscillations
P. S. Tambade

Stable equilibrium
If a slight displacement of particle from its equilibrium position
results only in small bounded motion about the point of
equilibrium, then it is said to be in stable equilibrium
Equilibrium
position
C

5. Oscillations
P. S. Tambade

Potential energy curve for stable equilibrium
-a +a
-x x
0
V(x)
x
Slope =
dV
dx
Tangent at A
A
Positive
F =
dV
dx
Force
F
Force is negative i.e. directed towards equilibrium
position
B
Tangent at B
Slope =
dV
dx
Negative
Force is positive i.e. directed towards equilibrium
position
F
Simulation
C

6. Oscillations
P. S. Tambade

Unstable equilibrium
If a slight displacement of the particle from its equilibrium position
results unbounded motion away from the equilibrium
position, then it is said to be in unstable equilibrium
Equilibrium
position
C

7. Oscillations
P. S. Tambade

Potential energy curve for unstable equilibrium
-a +a
-x x
0
V(x)
x
Slope =
dV
dx
Tangent at A
A
Negative
F =
dV
dx
Force
F
Force is positive i.e. directed away from equilibrium
position
B
Tangent at B
Slope =
dV
dx
Positive
Force is negative i.e. directed away from equilibrium
position
F
C

8. Oscillations
P. S. Tambade

Oscillatory Motion
Any motion that repeats itself after equal intervals of time is called
periodic motion.
If an object in periodic motion moves back and forth over the
same path, the motion is called oscillatory or vibratory motion
C

9. Oscillations
P. S. Tambade

Spring-Mass system
m
m
x= 0
x
–x
Relaxed mode
Extended mode
Compressed mode
F
F
m
We know that for an ideal
spring, the force is related
to the displacement bykxF
C

10. Oscillations
P. S. Tambade

Simple Harmonic MotionkxF
Linear simple harmonic motion :
Whentheforceactingontheparticleisdirectly
proportionaltothedisplacementandoppositein
direction,themotionissaidtobelinearsimpleharmonic
motion
Differential equation of motion is
m
d
2
x
dt
2
+ kx = 0
d
2
x
dt
2
+ ω
2
x = 0m
k
mk
2
where
Solution is
x= asin (ωt + )
(ωt + )is called phase and is called epoch of SHM
C

11. Oscillations
P. S. Tambade

•aand are determined uniquely by the position
and velocityof the particle at t= 0
•If at t= 0the particle is at x= 0, then = 0
•If at t= 0the particle is at x= a, then = π/2
•The phase of the motionis the quantity (ωt+ )
•x(t)is periodicand its value is the same each
time ωtincreases by 2πradians
x= asin (ωt + )
The displacement of particle from equilibrium position is
C

12. Oscillations
P. S. Tambade

Simpleharmonicmotion(orSHM)isthe
sinusoidalmotionexecutedbyaparticleof
massmsubjecttoone-dimensionalnet
forcethatisproportionaltothe
displacement oftheparticlefrom
equilibriumbutoppositeinsign
C

13. Oscillations
P. S. Tambade

x= asin (ωt + )
Equation of SHM is
The velocity is
v=
dx
dt
v = aωcos (ωt + )
or v = ω2
xa
2
The velocity is zero at extreme positions and maximum
at equilibrium position
C

14. Oscillations
P. S. Tambade

Graphs of Displacement and Velocity
x
t, time
T
T
ωt
π 2π3π
2
5π
2
π
2
3π 7π
2
4π
π
2
π
2
v
For =
π
2
x= asin (ωt + )
v = aωcos (ωt + )
The phase difference between velocity and displacement is
π
2
+a
-a
ωT
ωT
ωT = 2, The period of oscillation is T = 2/ ωT is called periodic time
C

15. Oscillations
P. S. Tambade

Periodic Time 2
T k
m
2T
The period of SHM is defined as the time taken by the
oscillator to perform one complete oscillation
After every time T, the particle will have the same
position, velocity and the directionttanconsiswhenT
ttanconsiswhenT
m
k
1
km
T
m
C

16. Oscillations
P. S. Tambade

The frequency represents the number of
oscillations that the particle undergoes per
unit time interval
•The inverse of the period is called the
frequency1
ƒ
2T
•Units are cycles per second = hertz (Hz)m
k
f
2
1
Frequency
C

17. Oscillations
P. S. Tambade

•The frequency and the period depend only on the mass of
the particle and the force constant of the spring
•They do not depend on the parameters of motion like
amplitude of oscillation
•The frequency is larger for a stiffer spring (large values of k)
and decreases with increasing mass of the particlek
m
2T m
k
f
2
1
C

18. Oscillations
P. S. Tambade

Displacement and acceleration
x
ωt
π 2π3π
2
5π
2
π
2
3π 7π
2
4π
π
A
x
For =
π
2
x= asin (ωt + ) A = -aω
2
sin (ωt + )
The phase difference between acceleration and displacement is π
π
C

19. Oscillations
P. S. Tambade

Energy
The potential energy is
V = k x
2
1
2
The kinetic energy is
K = mv
2
1
2
or K = mω
2
(a
2
–x
2
)
1
2
The total energy is
E = K + V
or E = mω
2
a
2
1
2
Thus, total energy of the oscillator is constant and proportional to
the square of amplitude of oscillations
C

20. Oscillations
P. S. Tambade

t x v K. E. P. E. E
0 + a 0 0
1
2
m ω
2
a
2

1
2
m ω
2
a
2

T/4 0 – a ω
1
2
m ω
2
a
2
0
1
2
m ω
2
a
2

T/2 – a 0 0
1
2
m ω
2
a
2

1
2
m ω
2
a
2

3T/4 0 + a ω
1
2
m ω
2
a
2
0
1
2
m ω
2
a
2

T +a 0 0
1
2
m ω
2
a
2

1
2
m ω
2
a
2

x
-a 0 +a
A
max
a
Summary …….
C
a
a
v
max
v
max

21. Oscillations
P. S. Tambade

-a +a
x
0
P. E.
K. E.
P. E. =K. E.
a/ 2-a/ 2
Energy
Graphical Representation of K. E. and P. E.
E = mω
2
a
2
1
2
The total mechanical energy is constant
The total mechanical energy is proportional to the square of the amplitude
Energy is continuously being transferred between potential energy stored
in the spring and the kinetic energy of the block
C

22. Oscillations
P. S. Tambade

Variation of K.E. and P. E. With time
x
ωt
π 2π3π
2
5π
2
π
2
3π 7π
2
4π
For =
π
2
x= asin (ωt + )
ωt
0
E
V = k x
2
1
2
K = mω
2
(a
2
–x
2
)
1
2
For one cycle of oscillation of particle there are two cycles for K. E.
and P.E.. Thus frequency of K. E. or P. E. is 2
P. E.
K. E.
0
C

23. Oscillations
P. S. Tambade

Angular SHM
If path of particle of a body performing an oscillatory
motion is curved, the motion is known as angular
simple harmonic motion
Definition:Angularsimpleharmonicmotionisdefined
astheoscillatorymotionofabodyinwhichthebodyis
acteduponbyarestoringtorque(couple)whichis
directlyproportionaltoitsangulardisplacementfrom
theequilibriumpositionanddirectedoppositetothe
angulardisplacement

24. Oscillations
P. S. Tambade
 2
2
dt
d
I 2
2
dt
d
I 0
2
2
Idt
d 0
2
2
2
dt
d )(tsin
0
is the torsion constantof the support wire
The restoring torque is
Newton’s Second Law gives
Angular SHM ……
I–moment of inertia

25. Oscillations
P. S. Tambade

•The torque equation produces a motion equation
for simple harmonic motion
•The angular frequency is
•The period is
–No small-angle restriction is necessary
–Assumes the elastic limit of the wire is not exceededI 2
I
T
C

26. Oscillations
P. S. Tambade

Simple Pendulum
•The equation of motion is

27. Oscillations
P. S. Tambade

•When angle is very small, we have sin +
g
l
=0
d
2
dt
2 T = 2
l
g
The Period is

28. Oscillations
P. S. Tambade

When but
then period is
When then period is T = T 1 + 1
4 2
2
sin 64
9
+ 4
2
sin + .... T= T1 +
2
16
But when angular arc is not
small ,then we have to
solve

29. Oscillations
P. S. Tambade

Dr. P. S. Tambadereceived an outstanding paper award in E-Learn
2008, Las Vegas