# Oscillatory Motion

The motion of the pendulum is the example of oscillatory motion. In the pendulum watch, its pendulum moves to and fro over a fixed potion which is the equilibrium potion. If the ideal conditions prevail oscillatory motion of any object will never end as in the ideal condition, friction due to air does not present.

## What is Oscillatory Motion?

When any object moves over a point repetitively then this type of motion of the object is the Oscillatory Motion. In the complete vacuum, the ideal condition can be approached as air will not be there to stop the object in oscillatory motion friction.

In the mechanical world, the vibration of strings and movement of spring are also come under oscillatory motion and are same as mechanical vibration. The oscillatory motion should not be confused with periodic motion. Objects in the periodic motions repeat its motion after a fixed duration or period of time whereas, in the oscillatory motion, the objects repeat their movement over a fixed position.

### Types of Oscillatory Motion

There are two types of oscillatory motions, namely, Linear Oscillatory Motion and Circular Oscillatory Motion.

In linear motion, the object moves left and right or up and down. Some examples of this type of linear motion are as below:

1. The vibration of strings of the musical instruments,
2. Movement of the fluid in a U-tube column and
3. Floating of ships or big vessels in the sea.

In the circular motion, though the object moves left to right but in circular form. Some examples of this type of motion are as below:

1. The motion of the solid sphere in a half hollow sphere
2. The motion of the pendulum in watch
3. A stringed object suspended on a nail
4. Motion of swing
5. The motion of a wheel

The equilibrium position of the oscillatory motion is the position about which the oscillations occur and in each oscillation, it is mandatory for the oscillating object to pass through this point. After some time when oscillatory motion stops due to friction of medium in which the object is oscillating, the oscillating object comes to rest on this point.

Objects, which are oscillating, can be said that they are vibrating. There is no significant difference between these two terms. But in general, when the object moves at high frequencies then the object is vibrating and when the movement takes place at low frequencies it is said that the object is oscillating.

The oscillatory motion should not be confused with periodic motion. Each oscillatory motion is always a periodic motion but it is not necessary that each periodic motion is oscillatory. A periodic motion may or may not be an oscillatory motion. For example, the motion of the car wheel is periodic but it is not oscillatory whereas the motion of stringed object nailed at the wall is both periodic as well as oscillatory.

### Simple Harmonic Motion (SHM) – An Oscillatory Motion

A simple form of oscillatory motion is Simple Harmonic Motion (SHM). In this motion, the restoring force is directly proportional to its displacement from its equilibrium position. This is Hooke’s Law.

Hooke’s law can be explained with the help of the following example. Suppose a block of mass m is tied up with long spring (with spring constant k) at one end and its other end is fixed on the wall. When the block moves away from the wall horizontally, spring tries to restore and due to this oscillatory motion starts. The restoring force which tries to restore the deformation of spring can be expressed by Hooke’s law which is,

$$F_{s} = – k x$$, the negative sign shows that the force is acting against the displacement x.

When the block is displaced to a position, x = A, and released, the restoring force will try to regain its equilibrium position. But when black reaches at its equilibrium position (x = 0) its potential energy converts to kinetic energy and this kinetic energy overshoots the block to another side of the equilibrium point until x = – A. from this point, block moves to right again and takes the position x = A. this way the motion continues until the friction force starts acting to stop the block. This frictional force equals to the restoring force. Thus, from Newton’s second law frictional force is

$$F_{r} = m a$$, where m is the mass of the block and ‘a’ is the acceleration of the block during motion.

$$F_{s} = F_{r}$$

Or

m a = – k x, it can be written in terms of x as

$$m\frac{\mathrm{d^{2}}x }{\mathrm{d} t^{2}}= – k x$$

or

$$m\frac{\mathrm{d^{2}}x }{\mathrm{d} t^{2}} + k x = 0$$

or

$$\frac{\mathrm{d^{2}}x }{\mathrm{d} t^{2}} + \omega ^{2}x = 0$$

Where, $$\omega =\sqrt{\frac{k}{m}}$$, which is the natural angular frequency of the system. The general solution of the above equation is

$$x(t)=A_{1}\cos \omega t+A_{2}\sin \omega t$$

Assuming,

$$A_{1}=D\sin\alpha$$ and $$A_{2}=D\cos\alpha$$

Then the above equation will become

$$x(t)= D\sin\alpha \cos \omega t+ D\cos\alpha \sin \omega t$$

Using the trigonometrical identity, sin (A+B)=sinAcosB+sinBcosA, we get

$$x(t)= D\sin(\alpha +\omega t)$$

So, it can be said that the oscillatory as well as periodic motion can be represented in terms of sine and cosine functions. Where D and $$\alpha$$ are constant and are equal to

$$D=\sqrt{A_{1}^{2}+A_{2}^{2}}$$

And $$\alpha=\tan^{-1}\left ( \frac{A_{1}}{A_{2}} \right )$$

And the period of this function is $$\frac{2\pi }{\omega }$$

## FAQs on the Oscillatory Motion

Q.1: What is an oscillatory motion and equilibrium point?

Answer: An oscillatory motion is a motion in which the object moves in such a way that it comes to and goes from a fixed point periodically. This fixed point is the equilibrium point. The equilibrium position of the oscillatory motion is the position about which the oscillations occur and in each oscillation, it is mandatory for the oscillating object to pass through this point. Example: electromagnetic waves, alternating current circuits, and molecular motion.

Q.2: How does oscillatory motion differ from periodic motion?

Answer: In the oscillatory motion an object moves over a fixed point again and again periodically. But in the periodic motion, the object moves from a fixed point after a certain period of time. In contrast, oscillatory motion can be oscillatory as well as periodic but the periodic motion cannot be an oscillatory motion. An example to explain it as the motion of the car wheel is periodic but it is not oscillatory whereas the motion of stringed object nailed at the wall is both periodic as well as oscillatory.

Q.3: What are the types of oscillatory motion?

Answer: There are two types of oscillatory motion.

1. Linear oscillatory motion and
2. Circular oscillatory motion

Q.4: What is Hooke’s law?

Answer: According to Hooke’s law, “the restoring force is directly proportional to the displacement from the equilibrium position”.

On a horizontal plane, a spring tied a block and fixed to the wall and block is put in such a way that it can move horizontally. When the block moves away from the wall horizontally, spring tries to restore and due to this oscillatory motion starts. The restoring force tries to stop the spring from deformation. This restoring force is given by

$$F_{s} = – k x$$, the negative sign shows that the force is acting against the displacement x.

This is Hooke’s law.

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