Periodic motion , Stable Equilibrium & Oscillation

Periodic Motion :
Any motion that repeats itself after equal interval of time is called periodic motion.

If a particle, undergoing periodic motion, covers the same path back and forth about a mean position, it is said to be executing an oscillation (or vibration).

Using advanced mathematics, it can be shown that the displacement of a particle in an oscillation can always be expressed in terms of sines and cosines.

This, coupled with the fact that the term harmonic is generally applied to expressions containing sine and cosine functions actuates us to use the term harmonic motion for the oscillation.

Every harmonic motion is characterised by:

(i) Time period

(ii) Frequency

(iii) Amplitude and

(iv) Phase

The Time Period (T) of a harmonic motion is the time required to complete one oscillation (or cycle).

The frequency (ν or n or f) of a harmonic motion is the number of oscillations per unit time.

From these definitions, it follows;

Our topic under discission is simple harmonic motion (SHM) and at this stage we just state it is a special type of harmonic motion.

Recalling Equilibrium :

Before investigating what exactly an SHM is, let us have a brief review of equilibrium of a particle (or body).

Without disturbing the continuity of discussion, it may be stated here that when a body is slightly disturbed from its stable equilibrium it may execute SHM.

This statement shows that equilibrium plays an important role in SHM and that’s why we are doing this review.

Meaning of Equilibrium :

For a particle or body,

If $\large \Sigma \vec{F} = 0 $ ; it is said to be in Translational Equilibrium.

If $\large \Sigma \vec{\tau} = 0 $ ; it is said to be in Rotational Equilibrium.

A rigid body is said to be in equilibrium only when it simultaneously confirms translational as well as rotational equilibrium. So, for showing a rigid body in equilibrium we have to show $\large \Sigma \vec{F} = 0 $ as well as $\large \Sigma \vec{\tau} = 0 $

Stable , Unstable & Neutral Equilibrium

Stable , Unstable and Neutral Equilibrium

Equilibrium can be further classified as stable, unstable and neutral equilibrium.

On being slightly disturbed from its equilibrium position, if a body

(i) tends to acquire the original configuration then the body is said to be in stable equilibrium.

(ii) tends to acquire a new position then the body is said to be in unstable equilibrium

(iii) remains at that position then the body is said to be in neutral equilibrium.

Stable, Unstable and neutral equilibrium in terms of potential energy

If potential energy of a body does not change with any change in its configuration then it is said to be in neutral equilibrium.

If potential energy of a body changes with change in its configuration then the body will have maximum potential energy at unstable equilibrium and minimum potential energy at stable equilibrium.

Stable Equilibrium and Oscillation :

Oscillation is intimately related with stable equilibrium.

To illustrate it, let us consider a typical curve between the position (x) of the particle and its potential energy (U) for a one dimensional particle motion in a conservative field.

Tangents drawn at B, C, D and E are parallel to the x-axis. This means, at these points, slope (dU/dx) is zero.

Recalling $\large F = -\frac{dU}{dx} $ , we can further say that at B , C , D and E , force acting on the particle is zero i.e. these are equilibrium positions.

For portions BC and DE, an increase in the value of x corresponds to an increase in the value of U.

The slope of the curve at any point in this portion is positive and hence , force( $\large F = -\frac{dU}{dx} $ ) is negative.

It means, in BC and DE region, the force acting on the particle tends to pull it in a region of lower potential energy.

Similarly it can be shown that for the portions AB and CD (where slope is negative and hence force is positive) again the force pulls the particle in the region of lower potential energy.

Thus any slight displacement of the particle, either way from the position of minimum potential energy results into a force tending to bring the particle back to its original position.

This force is often referred to as restoring force and site of minimum potential energy, as recalled earlier, is the position of stable equilibrium.

What happens if the particle is slightly disturbed from minimum potential energy position and then released ?

Let the particle be slightly displaced from B to a new position P and then released. Because of restoring force, it returns to B. Since the system is conservative, therefore the decrease in potential energy (= UP – UB) is compensated by an increase in kinetic energy. This kinetic energy at B influences the particle to go to other side of the curve.
Once again PE starts increasing which results into decrease in KE. At Q, kinetic energy is reduced to zero. The particle remains momentarily at rest and then starts moving back retracing its path under the action of restoring force.
Thus, on being slightly disturbed, the particle keeps on oscillating between the two extreme points P and Q. It remains confined in a bounded region (PBQ) and such a region always exists about a point of minimum potential energy or stable equilibrium.

Also Read :

→ S.H.M :Linear SHM & Angular SHM
→ Analytical Treatment to SHM
→ Kinetic Energy & Potential Energy & Total Energy in SHM
→ Average Value of P.E. & K.E. of Harmonic Oscillator
→ SHM as a projection of circular motion
→ Simple Pendulum in Inertial & Non Inertial Frame
→ Time period of a Long Pendulum
→ SHM of Spring Mass System
→ Physical Pendulum & Torsional Pendulum
→ Undamped & Damped simple harmonic oscillations

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