236
6 Oscillations
where A is the amplitude, (ωt + ε) is called the phase and ε is called the phase
difference.
The velocity v is given by
v = ±ω
A 2 − x 2
(6.7)
The acceleration is given by
a = −ω
2 x
(6.8)
The frequency of oscillation is given by
f =
ω
2π
=
1
2π
k
m
(6.9)
where m is the mass of the particle.
The time period is given by
T =
1
f
= 2π
m
k
(6.10)
Total energy (E) of the oscillator:
E = 1 / 2 m A
2
ω
2
(6.11)
K av = U av =
1
/ 4 m A
2
ω
2
(6.12)
Loaded spring:
T = 2π
M +
m
3
k
(6.13)
where M is the load and m is the mass of the spring.
If v 1 and v 2 are the velocities of a particle at x 1 and x 2 , respectively, then
T = 2π
x 2
2 − x 2
1
v 2
1 − v 2
2
(6.14)
A =
v 2
1 x 2
2 − v 2
2 x 2
1
v 2
1 − v 2
2
(6.15)