Chapter 6
Oscillations
Abstract Chapter 6 deals with simple harmonic motion and its application to various problems, physical pendulums, coupled systems of masses and springs, the
normal coordinates and damped vibrations.
6.1 Basic Concepts and Formulae
Simple Harmonic Motion (SHM)
In SHM the restoring force (F) is proportional to the displacement but is oppositely
directed.
F = −kx
(6.1)
where k is a constant, known as force constant or spring constant. The negative sign
in (6.1) implies that the force is opposite to the displacement.
When the mass is released, the force produces acceleration a given by
a = F/m = −k/m = −ω
2 x
(6.2)
where ω
2
= k/m
(6.3)
and ω = 2π f
(6.4)
is the angular frequency.
Differential equation for SHM:
d 2 x
dt 2 + ω
2 x = 0
(6.5)
Most general solution for (6.5) is
x = A sin(ωt + ε)
(6.6)
235
Oscillations
Abstract Chapter 6 deals with simple harmonic motion and its application to various problems, physical pendulums, coupled systems of masses and springs, the
normal coordinates and damped vibrations.
6.1 Basic Concepts and Formulae
Simple Harmonic Motion (SHM)
In SHM the restoring force (F) is proportional to the displacement but is oppositely
directed.
F = −kx
(6.1)
where k is a constant, known as force constant or spring constant. The negative sign
in (6.1) implies that the force is opposite to the displacement.
When the mass is released, the force produces acceleration a given by
a = F/m = −k/m = −ω
2 x
(6.2)
where ω
2
= k/m
(6.3)
and ω = 2π f
(6.4)
is the angular frequency.
Differential equation for SHM:
d 2 x
dt 2 + ω
2 x = 0
(6.5)
Most general solution for (6.5) is
x = A sin(ωt + ε)
(6.6)
235