A Classical Approach of Newtonian Mechanics

11 OSCILLATORY MOTION 11.2 Simple harmonic motion

0

!

executes simple harmonic motion about its equilibrium state. In physical terms,

if the initial displacement is positive (x > 0) then the restoring force overcom-

pensates, and sends the system past the equilibrium state (x = 0) to negative

displacement states (x < 0). The restoring force again overcompensates, and

sends the system back through x = 0 to positive displacement states. The motion

then repeats itself ad infinitum. The frequency of the oscillation is determined by

the spring stiffness, k, and the system inertia, m, via Eq. (11.5). In contrast, the
amplitude and phase angle of the oscillation are determined by the initial condi-

tions. Suppose that the instantaneous displacement and velocity of the mass at

t = 0 are x 0 and v 0 , respectively. It follows from Eq. (11.3) that

x 0 = x(t = 0) = a cos φ, (11.9)

v 0 = x ̇(t = 0 ) = a ω sin φ. (11.10)

Here, use has been made of the well-known identities cos(−θ) = cos θ and

sin(−θ) = − sin θ. Hence, we obtain

a =

q

x 2 + (v 0 /ω)^2 , (11.11)

and
φ = tan−^1

v 0

, (11.12)

ω x 0

since sin^2 θ + cos^2 θ = 1 and tan θ = sin θ/ cos θ.

The kinetic energy of the system is written

K =

1

m x ̇^2 =

2

m a^2 ω^2 sin^2 (ω t − φ)

2

. (11.13)

Recall, from Sect. 5.6, that the potential energy takes the form

1

U = k x^2 =

2

k a^2 cos^2 (ω t − φ)

2

. (11.14)

Hence, the total energy can be written

E = K + U =

a^2 k

2

, (11.15)