Oscillation
Physics
1 Basics
Definitions : Periodic motion refers to any motion that repeats itself at regular intervals of time.
Oscillation is a specific type of periodic motion that involves an object moving back and forth about a
central point. And SHM(Simple Harmonic Motion) is special type of oscillatory motion which moves
under a restoring force. Mathematically
∴ F ∝ −x
F = −kx
ma + kx = 0
d2 x
m + kx = 0
dt2
d2 x k
+ x=0
dt2 m
k
Now if we take ω 2 = m
d2 x
+ ω2 x = 0
dt2
This is known as the differential equation of SHM. To solve it we put x = Aeαt
∴ Aα2 eαt + Aω 2
α2 + ω 2 = 0
α = ±iω
Now the general solution is
∴ x = A1 eiωt + A2 e−iωt
x = A1 (cos ωt + i sin ωt) + A2 (cos ωt − i sin ωt)
x = (A1 + A2 ) cos ωt + (A1 − A2 )i sin ωt
x = A cos ωt + B sin ωt
From here we can get the phase amplitude form
x = A sin(ωt + ϕ) or x = A cos(ωt + δ)
1.1 Kinematics and Dynamics :
1.1.1 Velocity and Acceleration:
For velocity we differentiate the displacement equation
dx
∴ = Aω cos(ωt + ϕ)
dt
v = Aω cos(ωt + ϕ) (1)
Now for acceleration we differentiate equation (1)
dv
∴ = −Aω 2 sin(ωt + ϕ)
dt
1
, a = −Aω 2 sin(ωt + ϕ)
Now
vmax = Aω and amax = Aω 2
Hence
amax
ω=
vmax
1.1.2 Energy :
Kinetic energy for SHM
1 1 1 k 1
KE = mv 2 = mA2 ω 2 cos2 (ωt + ϕ) = mA2 cos2 (ωt + ϕ) = kA2 cos2 (ωt + ϕ)
2 2 2 m 2
Potential energy of SHM
1 2 1
PE = kx = kA2 sin2 (ωt + ϕ)
2 2
Total energy
∴ T E = P E + KE
1 1
= kA2 sin2 (ωt + ϕ) + kA2 cos2 (ωt + ϕ)
2 2
1 2 2
= kA sin (ωt + ϕ) + cos2 (ωt + ϕ)
2
1
= kA2
2
To simplify further
1
TE = mω 2 A2
2
1
= m × (2πf )2 A2
2
= 2mπ 2 f 2 A2
Thus
1
E ∝ A2 and E ∝ f 2 and E ∝
T2
Obtaining equations of motion from Total Energy Equation
1 2 1
∴ TE = kx + mv 2
2 2
d d 1 2 1 2
(T E) = kx + mv
dt dt 2 2
1 dx 1 dv
0 = k × 2x × + m × 2v ×
2 dt 2 dt
d2 x
kxv + mv 2 = 0
dt
d2 x k
vm( 2 + x) = 0
dt m
d2 x
+ ω2 x = 0
dt2
1.2 Superposition :
It is the phenomenon where a single particle is subjected to two or more harmonic oscillation simul-
taneously. Instead of choosing one path the particle follows a resultant path. Though it has some
limitation
1. Only applicable for small oscillation.
2. 2 SHM must have same nature.
2
Physics
1 Basics
Definitions : Periodic motion refers to any motion that repeats itself at regular intervals of time.
Oscillation is a specific type of periodic motion that involves an object moving back and forth about a
central point. And SHM(Simple Harmonic Motion) is special type of oscillatory motion which moves
under a restoring force. Mathematically
∴ F ∝ −x
F = −kx
ma + kx = 0
d2 x
m + kx = 0
dt2
d2 x k
+ x=0
dt2 m
k
Now if we take ω 2 = m
d2 x
+ ω2 x = 0
dt2
This is known as the differential equation of SHM. To solve it we put x = Aeαt
∴ Aα2 eαt + Aω 2
α2 + ω 2 = 0
α = ±iω
Now the general solution is
∴ x = A1 eiωt + A2 e−iωt
x = A1 (cos ωt + i sin ωt) + A2 (cos ωt − i sin ωt)
x = (A1 + A2 ) cos ωt + (A1 − A2 )i sin ωt
x = A cos ωt + B sin ωt
From here we can get the phase amplitude form
x = A sin(ωt + ϕ) or x = A cos(ωt + δ)
1.1 Kinematics and Dynamics :
1.1.1 Velocity and Acceleration:
For velocity we differentiate the displacement equation
dx
∴ = Aω cos(ωt + ϕ)
dt
v = Aω cos(ωt + ϕ) (1)
Now for acceleration we differentiate equation (1)
dv
∴ = −Aω 2 sin(ωt + ϕ)
dt
1
, a = −Aω 2 sin(ωt + ϕ)
Now
vmax = Aω and amax = Aω 2
Hence
amax
ω=
vmax
1.1.2 Energy :
Kinetic energy for SHM
1 1 1 k 1
KE = mv 2 = mA2 ω 2 cos2 (ωt + ϕ) = mA2 cos2 (ωt + ϕ) = kA2 cos2 (ωt + ϕ)
2 2 2 m 2
Potential energy of SHM
1 2 1
PE = kx = kA2 sin2 (ωt + ϕ)
2 2
Total energy
∴ T E = P E + KE
1 1
= kA2 sin2 (ωt + ϕ) + kA2 cos2 (ωt + ϕ)
2 2
1 2 2
= kA sin (ωt + ϕ) + cos2 (ωt + ϕ)
2
1
= kA2
2
To simplify further
1
TE = mω 2 A2
2
1
= m × (2πf )2 A2
2
= 2mπ 2 f 2 A2
Thus
1
E ∝ A2 and E ∝ f 2 and E ∝
T2
Obtaining equations of motion from Total Energy Equation
1 2 1
∴ TE = kx + mv 2
2 2
d d 1 2 1 2
(T E) = kx + mv
dt dt 2 2
1 dx 1 dv
0 = k × 2x × + m × 2v ×
2 dt 2 dt
d2 x
kxv + mv 2 = 0
dt
d2 x k
vm( 2 + x) = 0
dt m
d2 x
+ ω2 x = 0
dt2
1.2 Superposition :
It is the phenomenon where a single particle is subjected to two or more harmonic oscillation simul-
taneously. Instead of choosing one path the particle follows a resultant path. Though it has some
limitation
1. Only applicable for small oscillation.
2. 2 SHM must have same nature.
2
