Important Formulas: Oscillations | Physics Class 11 - NEET PDF Download

 Page 1


SIMPLE HARMONIC MOTION
S.H.M.
F = – kx
General equation of S.H.M. is x = A sin ( ?t + ?); ( ?t + ?) is phase of the
motion and ? is initial phase of the motion.
Angular Frequency ( ?) : ? = 
T
2 ?
 = 2 ?f
Time period (T) : T = 
?
? 2
= 
k
m
2 ?
m
k
Speed :
2 2
x A v ? ? ?
Acceleration : a = ? ? ?
2
x
Kinetic Energy (KE) :      
2
1
 mv
2
  = 
2
1
 m ?
2
 (A
2
 – x
2
) =
2
1
 k (A
2
 – x
2
)
Potential Energy (PE) :
2
1
 Kx
2
Total Mechanical Energy (TME)
= K.E. + P.E.  = 
2
1
 k (A
2
 – x
2
) + 
2
1
 Kx
2
 = 
2
1
 KA
2 
 (which is constant)
SPRING-MASS SYSTEM
(1)
smooth surface
?
 k
 m
? T = 2 ?
k
m
(2)
 T = 
K
2
?
? ? ? ? ?where ? = 
) m (m
m m
2 1
2 1
?
?known as reduced mass
Page 2


SIMPLE HARMONIC MOTION
S.H.M.
F = – kx
General equation of S.H.M. is x = A sin ( ?t + ?); ( ?t + ?) is phase of the
motion and ? is initial phase of the motion.
Angular Frequency ( ?) : ? = 
T
2 ?
 = 2 ?f
Time period (T) : T = 
?
? 2
= 
k
m
2 ?
m
k
Speed :
2 2
x A v ? ? ?
Acceleration : a = ? ? ?
2
x
Kinetic Energy (KE) :      
2
1
 mv
2
  = 
2
1
 m ?
2
 (A
2
 – x
2
) =
2
1
 k (A
2
 – x
2
)
Potential Energy (PE) :
2
1
 Kx
2
Total Mechanical Energy (TME)
= K.E. + P.E.  = 
2
1
 k (A
2
 – x
2
) + 
2
1
 Kx
2
 = 
2
1
 KA
2 
 (which is constant)
SPRING-MASS SYSTEM
(1)
smooth surface
?
 k
 m
? T = 2 ?
k
m
(2)
 T = 
K
2
?
? ? ? ? ?where ? = 
) m (m
m m
2 1
2 1
?
?known as reduced mass
COMBINATION OF SPRINGS
Series Combination : 1/k
eq
 = 1/k
1
 + 1/k
2
Parallel combination : k
eq
 = k
1
 + k
2
SIMPLE PENDULUM T = 2 ? 
g
?
  =  2 ?
. eff
g
?
 (in accelerating Refer-
ence Frame); g
eff
 is net acceleration due to pseudo force and gravitational
force.
COMPOUND PENDULUM / PHYSICAL PENDULUM
Time period (T) : T = 2 ?
? mg
?
where, ? = ?
CM
 + m ?
2
 ; ? is distance between point of suspension and
centre of mass.
TORSIONAL PENDULUM
Time period (T) : T = 2 ?
C
?
where, C = Torsional constant
Superposition of SHM’s along the same direction
x
1
 = A
1
 sin ?t  & x
2
 = A
2
 sin ( ?t + ?)
A
2
A
A
1
If equation of resultant SHM is taken as x = A sin ( ?t + ?)
A = 
? ? ? cos A A 2 A A
2 1
2
2
2
1
& tan ? = 
? ?
?
cos A A
sin A
2 1
2
1. Damped Oscillation
? Damping force
v b – F
?
?
?
? equation of motion is
dt
mdv
= –kx – bv
? b
2
 - 4mK > 0 over damping
Page 3


SIMPLE HARMONIC MOTION
S.H.M.
F = – kx
General equation of S.H.M. is x = A sin ( ?t + ?); ( ?t + ?) is phase of the
motion and ? is initial phase of the motion.
Angular Frequency ( ?) : ? = 
T
2 ?
 = 2 ?f
Time period (T) : T = 
?
? 2
= 
k
m
2 ?
m
k
Speed :
2 2
x A v ? ? ?
Acceleration : a = ? ? ?
2
x
Kinetic Energy (KE) :      
2
1
 mv
2
  = 
2
1
 m ?
2
 (A
2
 – x
2
) =
2
1
 k (A
2
 – x
2
)
Potential Energy (PE) :
2
1
 Kx
2
Total Mechanical Energy (TME)
= K.E. + P.E.  = 
2
1
 k (A
2
 – x
2
) + 
2
1
 Kx
2
 = 
2
1
 KA
2 
 (which is constant)
SPRING-MASS SYSTEM
(1)
smooth surface
?
 k
 m
? T = 2 ?
k
m
(2)
 T = 
K
2
?
? ? ? ? ?where ? = 
) m (m
m m
2 1
2 1
?
?known as reduced mass
COMBINATION OF SPRINGS
Series Combination : 1/k
eq
 = 1/k
1
 + 1/k
2
Parallel combination : k
eq
 = k
1
 + k
2
SIMPLE PENDULUM T = 2 ? 
g
?
  =  2 ?
. eff
g
?
 (in accelerating Refer-
ence Frame); g
eff
 is net acceleration due to pseudo force and gravitational
force.
COMPOUND PENDULUM / PHYSICAL PENDULUM
Time period (T) : T = 2 ?
? mg
?
where, ? = ?
CM
 + m ?
2
 ; ? is distance between point of suspension and
centre of mass.
TORSIONAL PENDULUM
Time period (T) : T = 2 ?
C
?
where, C = Torsional constant
Superposition of SHM’s along the same direction
x
1
 = A
1
 sin ?t  & x
2
 = A
2
 sin ( ?t + ?)
A
2
A
A
1
If equation of resultant SHM is taken as x = A sin ( ?t + ?)
A = 
? ? ? cos A A 2 A A
2 1
2
2
2
1
& tan ? = 
? ?
?
cos A A
sin A
2 1
2
1. Damped Oscillation
? Damping force
v b – F
?
?
?
? equation of motion is
dt
mdv
= –kx – bv
? b
2
 - 4mK > 0 over damping
? b
2
 - 4mK = 0 critical damping
? b
2
 - 4mK < 0 under damping
? For small damping the solution is of the form.
x = ? ?
m 2 / bt –
0
e A sin [ ?
1
t + ? ],  where   
2
m 2
b
–
m
k
' ?
?
?
?
?
?
?
?
?
?
?
?
? ?
For small b
? angular frequency  
0
, m / k ' ? ? ? ?
? Amplitude  
m 2
bt –
0
e A A ?
l
? Energy  E (t) =  
2
1
KA
2
 
m / bt –
e
? Quality factor or Q value , Q = 
| E |
E
2
?
?
 = 
Y
2
'
?
?
where  , 
2
2
m 4
b
.
m
k
' ? ?
    ,
m 2
b
Y
? ?
2. Forced Oscillations And Resonance
External Force F(t) = F
0
 cos ?
d
 t
x(t) = A cos ( ?
d
t + ?)
? ?
0
2
2 2 2 2 2
d d
F
A
m b
?
? ?
? ? ? ? ?
? ?
? ?
and   
0
d 0
v
tan
x
?
? ?
?
(a) Small Damping  
? ?
0
2 2
d
F
A
m
?
? ? ?
(b)  Driving Frequency Close to Natural Frequency  
0
d
F
A
b
?
?