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Page 1
K.V. Lumding; K.V. Karimganj; K.V. Langjing
187
Oscillations and Waves
? Periodic Motion: A motion which repeats itself over and over again after a
regular interval of time.
? Oscillatory Motion: A motion in which a body moves back and forth repeatedly
about a fixed point.
? Periodic function: A function that repeats its value at regular intervals of its
argument is called periodic function. The following sine and cosine functions are
periodic with period T.
f(t) = sin
and g(t) = cos
These are called Harmonic Functions.
Note :- All Harmonic functions are periodic but all periodic functions are
not harmonic.
One of the simplest periodic functions is given by
f(t) = A cos ?t [? = 2p/T]
If the argument of this function ?t is incre a se d b y a n int e g ral m u ltipl e o f 2 p rad ia n s, the value of the function remains the same. The function f(t) is then periodic and its
period, T is given by
T =
Thus the function f(t) is periodic with period T
f(t) = f(t +T)
Linear combination of sine and cosine functions
f(t) = A sin ?t + B cos ?t
A periodic function with same period T is given as
A = D cos ø and B = D sin ø
Page 2
K.V. Lumding; K.V. Karimganj; K.V. Langjing
187
Oscillations and Waves
? Periodic Motion: A motion which repeats itself over and over again after a
regular interval of time.
? Oscillatory Motion: A motion in which a body moves back and forth repeatedly
about a fixed point.
? Periodic function: A function that repeats its value at regular intervals of its
argument is called periodic function. The following sine and cosine functions are
periodic with period T.
f(t) = sin
and g(t) = cos
These are called Harmonic Functions.
Note :- All Harmonic functions are periodic but all periodic functions are
not harmonic.
One of the simplest periodic functions is given by
f(t) = A cos ?t [? = 2p/T]
If the argument of this function ?t is incre a se d b y a n int e g ral m u ltipl e o f 2 p rad ia n s, the value of the function remains the same. The function f(t) is then periodic and its
period, T is given by
T =
Thus the function f(t) is periodic with period T
f(t) = f(t +T)
Linear combination of sine and cosine functions
f(t) = A sin ?t + B cos ?t
A periodic function with same period T is given as
A = D cos ø and B = D sin ø
K.V. Lumding; K.V. Karimganj; K.V. Langjing
188
f(t) = D sin (?t + ø)
D = v
and ø =
? Simple Harmonic Motion (SHM): A particle is said to execute SHM if it moves
to and fro about a mean position under the action of a restoring force which is
directly proportional to its displacement from mean position and is always
directed towards mean position.
Restoring Force Displacement
F
W h e re ‘k’ is f o rce c o n s ta n t.
? Amplitude: Maximum displacement of oscillating particle from its mean position.
x
Max
=
? Time Period: Time taken to complete one oscillation.
? Frequency:
. Unit of frequency is Hertz (Hz).
1 Hz = 1
? Angular Frequency:
= 2 p?
S .I u n it ? = ra d
? Phase:
1. The Phase of Vibrating particle at any instant gives the state of the particle
as regards its position and the direction of motion at that instant.
It is denoted by ø.
2. Initial phase or epoch: The phase of particle corresponding to time t = 0.
It is denoted by ø.
? Displacement in SHM :
( ø
0
)
Where, = Displacement,
A = Amplitude
?t = Angular Frequency
ø
0
= Initial Phase.
Page 3
K.V. Lumding; K.V. Karimganj; K.V. Langjing
187
Oscillations and Waves
? Periodic Motion: A motion which repeats itself over and over again after a
regular interval of time.
? Oscillatory Motion: A motion in which a body moves back and forth repeatedly
about a fixed point.
? Periodic function: A function that repeats its value at regular intervals of its
argument is called periodic function. The following sine and cosine functions are
periodic with period T.
f(t) = sin
and g(t) = cos
These are called Harmonic Functions.
Note :- All Harmonic functions are periodic but all periodic functions are
not harmonic.
One of the simplest periodic functions is given by
f(t) = A cos ?t [? = 2p/T]
If the argument of this function ?t is incre a se d b y a n int e g ral m u ltipl e o f 2 p rad ia n s, the value of the function remains the same. The function f(t) is then periodic and its
period, T is given by
T =
Thus the function f(t) is periodic with period T
f(t) = f(t +T)
Linear combination of sine and cosine functions
f(t) = A sin ?t + B cos ?t
A periodic function with same period T is given as
A = D cos ø and B = D sin ø
K.V. Lumding; K.V. Karimganj; K.V. Langjing
188
f(t) = D sin (?t + ø)
D = v
and ø =
? Simple Harmonic Motion (SHM): A particle is said to execute SHM if it moves
to and fro about a mean position under the action of a restoring force which is
directly proportional to its displacement from mean position and is always
directed towards mean position.
Restoring Force Displacement
F
W h e re ‘k’ is f o rce c o n s ta n t.
? Amplitude: Maximum displacement of oscillating particle from its mean position.
x
Max
=
? Time Period: Time taken to complete one oscillation.
? Frequency:
. Unit of frequency is Hertz (Hz).
1 Hz = 1
? Angular Frequency:
= 2 p?
S .I u n it ? = ra d
? Phase:
1. The Phase of Vibrating particle at any instant gives the state of the particle
as regards its position and the direction of motion at that instant.
It is denoted by ø.
2. Initial phase or epoch: The phase of particle corresponding to time t = 0.
It is denoted by ø.
? Displacement in SHM :
( ø
0
)
Where, = Displacement,
A = Amplitude
?t = Angular Frequency
ø
0
= Initial Phase.
K.V. Lumding; K.V. Karimganj; K.V. Langjing
189
Case 1: When Particle is at mean position x = 0
v = v
=
v
max
= =
Case 2: When Particle is at extreme position x =
v = v
= 0
Acceleration
Case 3: When particle is at mean position x = 0,
acceleration =
( ) = 0.
Case 4: When particle is at extreme position then
acceleration =
? Formula Used :
1. ( ø
0
)
2. v =
v
, v
max
= ?A.
3.
?
2
( ø
0
)
a
max
= ?
2
A
4. Restoring force F = = m?
2
Where = force constant & ?
2
=
5. Angular freq. ? = 2 =
/
6. Time Period T = 2pv
= 2pv
7. Time Period T = 2pv
= 2pv
8. P .E a t d ispl a ce m e n t ‘y ’ f r o m m e a n p o sitio n
E
P
=
ky
2
=
m?
2
y
2
=
m?
2
A
2
sin
2
?t
Page 4
K.V. Lumding; K.V. Karimganj; K.V. Langjing
187
Oscillations and Waves
? Periodic Motion: A motion which repeats itself over and over again after a
regular interval of time.
? Oscillatory Motion: A motion in which a body moves back and forth repeatedly
about a fixed point.
? Periodic function: A function that repeats its value at regular intervals of its
argument is called periodic function. The following sine and cosine functions are
periodic with period T.
f(t) = sin
and g(t) = cos
These are called Harmonic Functions.
Note :- All Harmonic functions are periodic but all periodic functions are
not harmonic.
One of the simplest periodic functions is given by
f(t) = A cos ?t [? = 2p/T]
If the argument of this function ?t is incre a se d b y a n int e g ral m u ltipl e o f 2 p rad ia n s, the value of the function remains the same. The function f(t) is then periodic and its
period, T is given by
T =
Thus the function f(t) is periodic with period T
f(t) = f(t +T)
Linear combination of sine and cosine functions
f(t) = A sin ?t + B cos ?t
A periodic function with same period T is given as
A = D cos ø and B = D sin ø
K.V. Lumding; K.V. Karimganj; K.V. Langjing
188
f(t) = D sin (?t + ø)
D = v
and ø =
? Simple Harmonic Motion (SHM): A particle is said to execute SHM if it moves
to and fro about a mean position under the action of a restoring force which is
directly proportional to its displacement from mean position and is always
directed towards mean position.
Restoring Force Displacement
F
W h e re ‘k’ is f o rce c o n s ta n t.
? Amplitude: Maximum displacement of oscillating particle from its mean position.
x
Max
=
? Time Period: Time taken to complete one oscillation.
? Frequency:
. Unit of frequency is Hertz (Hz).
1 Hz = 1
? Angular Frequency:
= 2 p?
S .I u n it ? = ra d
? Phase:
1. The Phase of Vibrating particle at any instant gives the state of the particle
as regards its position and the direction of motion at that instant.
It is denoted by ø.
2. Initial phase or epoch: The phase of particle corresponding to time t = 0.
It is denoted by ø.
? Displacement in SHM :
( ø
0
)
Where, = Displacement,
A = Amplitude
?t = Angular Frequency
ø
0
= Initial Phase.
K.V. Lumding; K.V. Karimganj; K.V. Langjing
189
Case 1: When Particle is at mean position x = 0
v = v
=
v
max
= =
Case 2: When Particle is at extreme position x =
v = v
= 0
Acceleration
Case 3: When particle is at mean position x = 0,
acceleration =
( ) = 0.
Case 4: When particle is at extreme position then
acceleration =
? Formula Used :
1. ( ø
0
)
2. v =
v
, v
max
= ?A.
3.
?
2
( ø
0
)
a
max
= ?
2
A
4. Restoring force F = = m?
2
Where = force constant & ?
2
=
5. Angular freq. ? = 2 =
/
6. Time Period T = 2pv
= 2pv
7. Time Period T = 2pv
= 2pv
8. P .E a t d ispl a ce m e n t ‘y ’ f r o m m e a n p o sitio n
E
P
=
ky
2
=
m?
2
y
2
=
m?
2
A
2
sin
2
?t
K.V. Lumding; K.V. Karimganj; K.V. Langjing
190
9. K .E . a t d ispla ce m e n t ‘ y ’ f rom t h e m e a n p o siti o n
E
K
=
k(
) =
m?
2
(A
2
– y
2
)
=
m?
2
A
2
cos
2
?t
10. Total Energy at any point
E
T
=
kA
2
=
m?
2
A
2
= 2p
2
mA
2
?
2
11. Spring Factor K = F/y
12. P e ri o d O f o scil lat io n o f a m a ss ‘ m ’ su s p e n d e d f ro m a m a ssl e ss sp ri n g o f f o rce co n sta n t ‘ k’
T = 2p v
For two springs of spring factors k
1
and k
2
connected in parallel effective spring
factor
k = k
1
+ k
2
T=2p
v
+
13. For t w o sp ri n g s co n n e cte d in se ri e s, e ff e c tiv e sp ri n g f a cto r ‘k’ is g iv e n a s
Or
+
T=2 v
(
+
)
Note:- When length of a spring is made ‘n’ times its spring factor
becomes
times and hence time period increases v times.
14. W h e n sp ri n g is cu t in to ‘n ’ e q u a l p iec e s, s p ri n g f a ct o r o f e a c h p a rt b e co m e s
‘nk’.
v
15. Oscillation of simple pendulum
v
/
v
/
16. For a liq u id o f d e n sity ? co n t a ine d in a U -tube up to h e ig h t ‘h’
Page 5
K.V. Lumding; K.V. Karimganj; K.V. Langjing
187
Oscillations and Waves
? Periodic Motion: A motion which repeats itself over and over again after a
regular interval of time.
? Oscillatory Motion: A motion in which a body moves back and forth repeatedly
about a fixed point.
? Periodic function: A function that repeats its value at regular intervals of its
argument is called periodic function. The following sine and cosine functions are
periodic with period T.
f(t) = sin
and g(t) = cos
These are called Harmonic Functions.
Note :- All Harmonic functions are periodic but all periodic functions are
not harmonic.
One of the simplest periodic functions is given by
f(t) = A cos ?t [? = 2p/T]
If the argument of this function ?t is incre a se d b y a n int e g ral m u ltipl e o f 2 p rad ia n s, the value of the function remains the same. The function f(t) is then periodic and its
period, T is given by
T =
Thus the function f(t) is periodic with period T
f(t) = f(t +T)
Linear combination of sine and cosine functions
f(t) = A sin ?t + B cos ?t
A periodic function with same period T is given as
A = D cos ø and B = D sin ø
K.V. Lumding; K.V. Karimganj; K.V. Langjing
188
f(t) = D sin (?t + ø)
D = v
and ø =
? Simple Harmonic Motion (SHM): A particle is said to execute SHM if it moves
to and fro about a mean position under the action of a restoring force which is
directly proportional to its displacement from mean position and is always
directed towards mean position.
Restoring Force Displacement
F
W h e re ‘k’ is f o rce c o n s ta n t.
? Amplitude: Maximum displacement of oscillating particle from its mean position.
x
Max
=
? Time Period: Time taken to complete one oscillation.
? Frequency:
. Unit of frequency is Hertz (Hz).
1 Hz = 1
? Angular Frequency:
= 2 p?
S .I u n it ? = ra d
? Phase:
1. The Phase of Vibrating particle at any instant gives the state of the particle
as regards its position and the direction of motion at that instant.
It is denoted by ø.
2. Initial phase or epoch: The phase of particle corresponding to time t = 0.
It is denoted by ø.
? Displacement in SHM :
( ø
0
)
Where, = Displacement,
A = Amplitude
?t = Angular Frequency
ø
0
= Initial Phase.
K.V. Lumding; K.V. Karimganj; K.V. Langjing
189
Case 1: When Particle is at mean position x = 0
v = v
=
v
max
= =
Case 2: When Particle is at extreme position x =
v = v
= 0
Acceleration
Case 3: When particle is at mean position x = 0,
acceleration =
( ) = 0.
Case 4: When particle is at extreme position then
acceleration =
? Formula Used :
1. ( ø
0
)
2. v =
v
, v
max
= ?A.
3.
?
2
( ø
0
)
a
max
= ?
2
A
4. Restoring force F = = m?
2
Where = force constant & ?
2
=
5. Angular freq. ? = 2 =
/
6. Time Period T = 2pv
= 2pv
7. Time Period T = 2pv
= 2pv
8. P .E a t d ispl a ce m e n t ‘y ’ f r o m m e a n p o sitio n
E
P
=
ky
2
=
m?
2
y
2
=
m?
2
A
2
sin
2
?t
K.V. Lumding; K.V. Karimganj; K.V. Langjing
190
9. K .E . a t d ispla ce m e n t ‘ y ’ f rom t h e m e a n p o siti o n
E
K
=
k(
) =
m?
2
(A
2
– y
2
)
=
m?
2
A
2
cos
2
?t
10. Total Energy at any point
E
T
=
kA
2
=
m?
2
A
2
= 2p
2
mA
2
?
2
11. Spring Factor K = F/y
12. P e ri o d O f o scil lat io n o f a m a ss ‘ m ’ su s p e n d e d f ro m a m a ssl e ss sp ri n g o f f o rce co n sta n t ‘ k’
T = 2p v
For two springs of spring factors k
1
and k
2
connected in parallel effective spring
factor
k = k
1
+ k
2
T=2p
v
+
13. For t w o sp ri n g s co n n e cte d in se ri e s, e ff e c tiv e sp ri n g f a cto r ‘k’ is g iv e n a s
Or
+
T=2 v
(
+
)
Note:- When length of a spring is made ‘n’ times its spring factor
becomes
times and hence time period increases v times.
14. W h e n sp ri n g is cu t in to ‘n ’ e q u a l p iec e s, s p ri n g f a ct o r o f e a c h p a rt b e co m e s
‘nk’.
v
15. Oscillation of simple pendulum
v
/
v
/
16. For a liq u id o f d e n sity ? co n t a ine d in a U -tube up to h e ig h t ‘h’
K.V. Lumding; K.V. Karimganj; K.V. Langjing
191
v
/
17. For a body dropped in a tunnel along the diameter of earth
v
/ , where R = Radius of earth
18. Resonance: If the frequency of driving force is equal to the natural frequency
of the oscillator itself, the amplitude of oscillation is very large then such
oscillations are called resonant oscillations and phenomenon is called
resonance.
Waves
Angular wave number: It is phase change per unit distance.
i.e.
?
? 2
? k
, S.I unit of k is radian per meter.
Relation between velocity, frequency and wavelength is given as :-
?? ? V
Velocity of Transverse wave:-
(i )In so li d mole cu les h a v ing m o d u lus o f ri g idity ‘ ? ’ a n d d e n sity ‘?’ is ?
?
? V
(i i) In st ri n g f o r m a ss p e r un it le n g th ’m ’ a n d te n sion ‘ T ’ is
m
T
V ?
Velocity of longitudinal wave:-
(i) in solid
?
Y
V ? , Y = y o u n g ’ s m o d u lus
(ii) in liquid
?
K
V ? , K= bulk modulus
Read More
FAQs on Oscillations & Waves - Detailed Notes - Class 11
| 1. What are oscillations and waves? |  |
Ans. Oscillations refer to the repetitive motion of an object or a system around an equilibrium position. Waves, on the other hand, are the disturbances that propagate through a medium or space. They carry energy from one point to another without transferring matter.
| 2. What is the difference between mechanical waves and electromagnetic waves? | |
Ans. Mechanical waves require a medium (such as air, water, or solids) to propagate, while electromagnetic waves can travel through a vacuum as they do not require a medium. Mechanical waves include sound waves, water waves, and seismic waves, while electromagnetic waves include light waves, radio waves, and X-rays.
| 3. How are oscillations and waves related to each other? | |
Ans. Oscillations are the source of waves. When an object or a system undergoes repeated back-and-forth or up-and-down motion, it creates a disturbance that propagates as a wave. The motion of particles in the medium causes the wave to transfer energy from one point to another.
| 4. What are the different types of waves? | |
Ans. There are several types of waves, including transverse waves, longitudinal waves, and surface waves. Transverse waves have the particles of the medium oscillating perpendicular to the direction of wave propagation. Longitudinal waves have the particles oscillating parallel to the direction of wave propagation. Surface waves are a combination of both transverse and longitudinal waves and occur at the interface between two different mediums.
| 5. How do waves exhibit properties like reflection, refraction, and diffraction? | |
Ans. Waves exhibit reflection when they bounce back from a surface, refraction when they change direction upon passing through different mediums, and diffraction when they bend around obstacles or pass through small openings. These phenomena occur due to the interaction of waves with the boundaries or irregularities in the medium they propagate through. These properties are fundamental to the behavior of waves in various applications such as optics and acoustics.
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