TIME PERIOD CALCULATION
dx
2
+
ω
=0
x
2
dt
2
- Displacement - x = A sin (ωt + φ)
- Velocity -V = dx = ω A Cos(ωt + φ)
dt
- Acceleration - a = A sin(wt + ) =
K → spring Constant
-ω2x
Graph of a - t
Graph of X - t
A
t
0
–A
Graph of v - t
T
2
2 ω A
T t
(i) keq = K1 + K2
T
k1
ENERGY OF LINEAR S.H.M
T = 2π
k1
k2
-A
A
0.8
1.0
1.2
1.4
d2 θ 2
(i) Different Equation → 2 tω θ = 0
dt
⇒ Displacement → θ = θo sin (ωt + S)
⇒ Torque → T = Kθ
K
−Kθ
; Angular accelartion → ∝ =
I
1
I
K
Physical Pendulum
:- Time period → T = 2π
I : MoI of system
M : Mass of System
I
mgd
θ
mg sinθ
d: distance between com and hinge
m
F ∝ -θ;
F = -Kθ;
Time period → = 2π

g
mg
Torsional Pendulum
T∝θ
T = -Cθ [C = Torsional Constant]
Time period – T = 2π
I
C
I : Moment of Inertia
m1m2
µ
T = 2π
= 2π
K (m1 + m2 )
k
P.E.
P.E.
m2
ET
k
Kmax or Umax or
X
m1
1
K A 2 cos2 (ωt + φ)
2
K.E
K, U
1
K (A 2 − x 2 )
2
1
→ P.E → U = K A 2 sin2 (ωt + φ)
2
K.E → K =
K.E → K =
P.E
0.6
oscillator as a function of the angular
frequency of the driving force
ANGULAR S.H.M
Simple Pendulum
m1m2
Reduced Mass: µ =
m1 + m2
T.E
2
3
ω → Natural Frequency
m(k1 + k 2 )
K1 K 2
k2
1
ωd → Driving Frequency
m
k eq
TWO BLOCKS SPRING SYSTEM
Energy
1 2
Kx
2
(2) Amplitude → A1 = Fo/wdb
K1 K 2
;
K1 + K 2
T = 2π
t
Fo
m(w 2 − w 2d )
⇒ Time period – T = 2π
m
→ P.E → U =
(1) Amplitude (For → wd >> ω) → A1 =
⇒ Angular Velocity → W =
(ii) Keq =
m
T = 2π
;
k eq
m
FORCED OSCILLATION
m
k eq
Time Period → T = 2π
m
T = 2π
k1 + k 2
T
2
velocity (v)

k 
m
2π
= 2π
ω
=

 Time period T =
k
m
ω

k
b2
,
−
m 4m2
Where – b = damping
Constant
Spring Block System
acceleration (a)
Displacement
X



(1) Force → F = − mω2x or F = − k x ;
(2) Angular Frequency → w1 =
PENDULUM
- Differential Equation of S.H.M
(1) Amplitude → A1 = Ae-bt/2m
Amplitude
CHARACTERISTICS OF LINEAR SHM
DAMPED AND FORCE OSCILLATIONS
SIMPLE HARMONIC MOTION
DAMPED OSCILLATION
ωt
anand_mani16
DR. Anand Mani
https://www.anandmani.com/
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