## Harmonic Motion

Any motion, which represents itself in equal intervals of time is called ** periodic motion**. Often, the displacement of a particle in periodic motion can be expressed in terms of sine and cosine functions. Since the term

*harmonic*is applied to expressions containing these functions, periodic motion is also called

**.**

*harmonic motion*If a particle in periodic motion moves back and *forth *(or *to *and *fro*) over the same path, then its motion is called ** oscillatory **or

**. The examples of**

*vibratory**oscillatory*or

*vibratory*motion are:

– the motion of a pendulum

– the motion of a spring fixed at one end, which is stretched or compressed and then released

– the motion of a violin string

– the motion of atoms in molecules or in a solid lattice

– the motion of air molecules as a sound wave passes by

** Characteristics of a Harmonic Motion **

The basic quantities characterizing a periodic motion are the *amplitude*, *period *and *frequency *of vibrations.

** Amplitude ( A)**

The amplitude of oscillations is the maximum displacement of a vibrating body from the position of equilibrium.

** Time Period ( T)**

The time period of oscillations is defined as the time between two successive identical positions passed by the body in the same direction.

**Frequency ( f) **

The frequency of oscillations is the number of cycles of vibrations of a body completed in one second. The frequency is related to the time period as

The *SI *unit of frequency is s^{-1} or Hz (hertz)

In mechanical oscillations a body oscillates about its *mean position *which is also its ** equilibrium position**. At the equilibrium position

*no*net force (or torque) acts on the oscillating body. The

*displacement*(linear or angular) of an oscillating particle is its distance (linear or angular) from the equilibrium position at any instant.

When a body oscillates along a straight line within two fixed limits; its *displacement x* changes periodically in both

*magnitude*and

*direction*; its

*velocity*and

**v***acceleration*also vary periodically in magnitude and direction, shown in figure. Since

**a****=**

*F**m*, therefore, the force acting on the body also varies in

**a***magnitude*and

*direction*with time.

In terms of energy, we can say that a particle executing harmonic motion moves *back *and *forth *about a point at which the *potential energy *is *minimum *(equilibrium position). The force acting on the body at any position is given by

When a body is displaced from its equilibrium it is acted upon by a *restoring *force (or torque) which always acts to accelerate the body in the direction of its equilibrium position as shown in the figure.

**DRILL EXERCISE****-1**

- A particle of mass
*m*is executing oscillation about the origin on the*x*-axis. Its potential energy is*U*(*x*) =*k*|*x*|^{3}, where*k*is a positive constant. If the amplitude of oscillation is*a*, then its time period*T*is

** **(a) proportional to 1÷√a (b) independent of *a*

** **(c) proportional to √a (d) proportional to *a*^{3/2}

** **

- A particle free to move along the
*x*-axis has potential energy given by*U*(*x*) =*k*[1 – exp (–*x*^{2})] for –£*x*£ + , where*k*is a positive constant of appropriate dimensions. Then

(a) at points away from the origin, the particle is in unstable equilibrium

(b) for any finite non-zero value of *x*, there is a force directed away from the origin

** **(c) if its total mechanical energy is it has its minimum kinetic energy at the origin

(d) for small displacements from *x* = 0, the motion is simple harmonic

**SIMPLE HARMONIC MOTION **

Let us consider an oscillatory particle along a straight line whose potential energy function varies as

where *k* is a constant

The force acting on the particle is given by
Such an oscillatory motion in which restoring The potential energy function of such a particle is represented by a symmetric curve as shown in the figure |

** Note **that the limits of oscillation are equally spaced about the equilibrium position. This is *not *true for the motion shown in figure**(1)**, which is harmonic, but *not *simple harmonic.

Applying Newton’s Second Law in equation **(4)**, we get

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