simple harmonic motion by Sachin jangid

1. SIMPLE HARMONIC MOTION
B.TECH-CSE
SECTION –A
SEM. - 𝑰 𝒔𝒕
SUBMITTED BY:-
SACHIN JANGID
SUBMITTED TO:-
Mr. UMESH KUMAR
DWIVEDI

2. CONTENTS:
1. Periodic motion
2. Simple harmonic motion
3. Amplitude
4. Phase
5. Angular frequency
6. Period
7. Velocity of simple harmonic motion
8. Acceleration of simple harmonic motion
9. Energy in simple harmonic motion
10. Damped simple harmonic motion
11. Forced oscillations and resonance

3. Periodic motion
• Periodic (harmonic) motion – self-repeating
motion
• Oscillation – periodic motion in certain direction
• Period (T) – a time duration of one oscillation
• Frequency (f) – the number of oscillations per unit
time, SI unit of frequency 1/s = Hz (Hertz)
T
f
1

Heinrich Hertz
(1857-1894)

4. Simple harmonic motion
Simple harmonic motion – motion that repeats itself
and the displacement is a sinusoidal function of time
)cos()(   tAtx

5. Amplitude
• Amplitude – the magnitude of the maximum
displacement (in either direction)
)cos()(   tAtx

6. Phase
)cos()(   tAtx

7. Phase constant
)cos()(   tAtx

8. Angular frequency
)cos()(   tAtx
0
)(coscos TtAtA  
)2cos(cos  
)(cos)2cos( Ttt  
T 2
T


2

f 2

9. Period
)cos()(   tAtx

2
T

10. Differential equation of SHM
A differential equation is simply an equation containing a
derivative. Since the motion is 1D, we can drop the vector
arrows and use sign to indicate direction.
The constants k and m and both positive, so the k/m is always
positive, always.For notational convenience, we write k/ m  2
.
(The square on the  reminds us that 2
is always positive.) The
differential equation becomes
Fnet  m a Fnet   k x
2
a  dv / dt  d2
x / dt2
and  m a   k x
d x  -kx/m

11. This is the differential equation for SHM. We seek a solution y= y(t)
to this equation, a function y = y(t) whose second time derivative is
the function y(t) multiplied by a negative constant (2
= k/m).
The way you solve differential equations is the same way you solve
integrals: you guess the solution and then check that the solution
works.
Based on observation, we guess a sinusoidal solution
x(t)  Acost  
where A,  are any constants and (as we'll show)   √k/m
d2
x / dt2
  2
x

12. Velocity of simple harmonic motion
)cos()(   tAtx
dt
tdx
tv
)(
)( 
dt
tAd )]cos([  

)sin()(   tAtv

13. Acceleration of simple harmonic motion
)cos()(   tAtx
2
2
)()(
)(
dt
txd
dt
tdv
ta 
)cos(2
  tA
)()( 2
txta 

14. The force law for simple harmonic
motion
• From the Newton’s Second Law:
•
•For simple harmonic motion, the force is
proportional to the displacement
• Hooke’s law:
2
mk 
kxF 
maF  xm 2

m
k

k
m
T 2

15. Energy in simple harmonic motion
• Potential energy of a spring:
• Kinetic energy of a mass:
2/)( 2
kxtU  )(cos)2/( 22
  tkA
2/)( 2
mvtK  )(sin)2/( 222
  tAm
)(sin)2/( 22
  tkA km 2


16. Energy in simple harmonic motion
 )()( tKtU
)(sin)2/()(cos)2/( 2222
  tkAtkA
 )(sin)(cos)2/( 222
  ttkA
)2/( 2
kA )2/( 2
kAKUE 

17. Pendulums
• Simple pendulum:
• Restoring torque:
• From the Newton’s Second Law:
• For small angles
)sin(  gFL
 I
 sin

I
mgL

)sin( gFL

18. Pendulums
• Simple pendulum:
• On the other hand
L
at


I
mgL

L
s
 s
I
mgL
a 
)()( 2
txta 
I
mgL


19. Pendulums
• Simple pendulum:
I
mgL
 2
mLI 
2
mL
mgL

L
g

g
L
T 


2
2


20. Simple harmonic motion and uniform
circular motion
• Simple harmonic motion is the projection of uniform
circular motion on the diameter of the circle in which
the circular motion occurs

21. Pendulums
• Physical pendulum:
I
mgh

mgh
I
T 


2
2


22. Simple harmonic motion and uniform
circular motion
• Simple harmonic motion is the projection of uniform
circular motion on the diameter of the circle in which
the circular motion occurs
)cos()(   tAtx
dt
tdx
tvx
)(
)( 
)sin()(   tAtvx

23. Simple harmonic motion and uniform
circular motion
• Simple harmonic motion is the projection of uniform
circular motion on the diameter of the circle in which
the circular motion occurs
dt
tdx
tvx
)(
)( 
)sin()(   tAtvx
)cos()(   tAtx

24. Simple harmonic motion and uniform
circular motion
• Simple harmonic motion is the projection of uniform
circular motion on the diameter of the circle in which
the circular motion occurs
2
2
)(
)(
dt
txd
tax 
)cos()(   tAtx
)cos()( 2
  tAtax

25. Where the force is proportional to the speed of the moving object and
acts in the direction opposite the motion.
The retarding force can be expressed as:
R = - bv ( where b is a constant called damping coefficient)
and the restoring force of the system is – kx,
then we can write Newton's second law as
  xxx mabvkxF 2
2
dt
xd
m
dt
dx
bkx 
When the retarding force is small compared with the max restoring force
that is, b is small the solution is,
)cos()( 2
 

tAetx
t
m
b
2
)
2
(
m
b
m
k

Damped simple harmonic motion

26. represent the position vs time for a
damped oscillation with decreasing
amplitude with time
The fig. shows the position as a function in time of the object oscillation in
the presence of a retarding force, the amplitude decreases in time, this
system is know as a damped oscillator. The dashed line which defined the
envelope of the oscillator curve, represent the exponential factor

27. as the value of "b" increase the amplitude of the oscillations
decreases more and more rapidly.
When b reaches a critical value bc ( ), the system does
not oscillate and is said to be critically damped.
And when the system is overdamped.
oc mb 2/
oc mb 2/
The fig. represent position versus
time:
•under damped oscillator
•critical damped oscillator
- Overdamped oscillator.

29. For the forced oscillator is a damped oscillator driven by an
external force that varies periodically
Where
Forced oscillations
where ω is the angular frequency of the driving force and Fo is
a constant
From the Newton's second law
tFtF o sin)( 
2
2
sin
dt
xd
mkx
dt
dx
btFmaF o  
)cos(   tAx
2
222
)(
/








m
b
mF
A
o
o



30. The last two equations show the driving force and the
amplitude of the oscillator which is constant for a
given driving force.
For small damping the amplitude is large when the
frequency of the driving force is near the natural
frequency of oscillation, or when ω͌ ≈ ωo the is called
the resonance and the natural frequency is called the
resonance frequency.
is the natural frequency of the un-
damped oscillator (b=0).m
k
o 

31. Amplitude versus the frequency, when the frequency of the
driving force equals the natural force of the oscillator,
resonance occurs. Note the depends of the curve as the value
of the damping coefficient b.