A Classical Approach of Newtonian Mechanics

11 OSCILLATORY MOTION 11.2 Simple harmonic motion

m

system is representative of the motion of a wide range of systems when they are

slightly disturbed from a stable equilibrium state.

Newton’s second law gives following equation of motion for the system:

m x ̈ = −k x. (11.2)

This differential equation is known as the simple harmonic equation, and its solu-

tion has been known for centuries. In fact, the solution is

x = a cos(ω t − φ), (11.3)

where a, ω, and φ are constants. We can demonstrate that Eq. (11.3) is in-

deed a solution of Eq. (11.2) by direct substitution. Substituting Eq. (11.3) into

Eq. (11.2), and recalling from calculus that d(cos θ)/dθ = − sin θ and d(sin θ)/dθ =

cos^ θ,^ we^ obtain^

— m ω^2 a cos(ω t − φ) = −k a cos(ω t − φ). (11.4)

It follows that Eq. (11.3) is the correct solution provided

ω =

‚

., k

. (11.5)

Figure 95 shows a graph of x versus t obtained from Eq. (11.3). The type

of motion shown here is called simple harmonic motion. It can be seen that the

displacement x oscillates between x = −a and x = +a. Here, a is termed the

amplitude of the oscillation. Moreover, the motion is periodic in time (i.e., it

repeats exactly after a certain time period has elapsed). In fact, the period is

2 π

T =. (11.6)

ω

This result is easily obtained from Eq. (11.3) by noting that cos θ is a periodic

function of θ with period 2 π. The frequency of the motion (i.e., the number of

oscillations completed per second) is

1

f = =

T

ω

. (11.7)
2 π