Free Forced Damped Oscillations

In this section, we will define oscillating motion which comes under the type of periodic movements. You will also come across detailed explanations on concepts like how to find periods of oscillation, simple harmonic motion, and different types of oscillatory motions, which will help you gain a better understanding of the subject. 

In our daily lives, we come across several such incidents that involve repetitive movements of objects. Be it the movement of your bicycle wheels, a football’s motion as you kick it, or strings of a playing guitar. If you observe closely, a lot of these movements take place in a rhythm while others do not. The former are grouped as periodic motions while the latter are categorized as non-periodic motions. 

What is Oscillation?

Oscillation is a periodic variation in time of a matter about its mean value or between two fixed states.  

One cycle of periodic to and fro motion of the body about its central position is one oscillation. Mechanical oscillations are called vibrations. A particle is vibrating means it oscillates between two points between its central point.

The height or the maximum distance that the oscillation takes place at is called the amplitude. It is measured in metres. While the time taken to complete one complete oscillation cycle is called its time period, the number of complete cycles that occur in one second is called the frequency. Frequency basically reciprocates the time period. The time period is calculated in seconds and the frequency is measured in hertz.

Properties of Oscillation

Here are the properties of oscillation:

• The frequency is defined as the number of complete oscillations per unit time.

• The amplitude is the maximum displacement of an oscillator from its equilibrium position.

• The time period is the time taken for one complete oscillation, in seconds. 

• The relationship of frequency and period is f = 1/T.

What are Some Examples of Oscillation?

There are some of the observed motions that serve as examples of oscillation.

Here are some of them.

• The motion of a pendulum bob in a clock. It is a repetitive motion with the bob rising to its right, coming back to its centre position, again rising to its left, and finally going back to its centre again. This total cycle constitutes 1 oscillation. 

• Movement of springs - the repetitive downward and upward motion.

• Tides in the sea. 

• The vibration of strings in a guitar and other string instruments.

• Elastic media. 

• A sine wave. 

Calculation of Oscillation

Several measurements define oscillation. Here, we have explained each parameter with their respective unit of measure and the formulae to calculate their value and also understand how to define oscillating motion.

• Period of Oscillation 

The time taken by an oscillating body to complete one cycle of motion is termed as its oscillation period. It is generally measured in second and is denoted by T.

Here’s how to calculate period of oscillation.

\[T = 2 \pi  (\sqrt{L}{g}) \]

Where L represents the length of a pendulum and g is the acceleration due to gravity.

• Frequency of Oscillation 

The number of oscillations a body can complete in one second is known as its frequency of oscillation. It is primarily expressed in the SI unit of Hertz and is denoted by the letter f.

You can calculate the value of f with its relation to period T of oscillation as follows:

\[f = \frac{1}{T}\]

• Amplitude of Oscillation 

The maximum amount of displacement of an oscillating body from its central position is known as its amplitude. Its value is measured in meter and it is denoted by A.

With a known value of A, displacement x is calculated as -

x = A cos 2πft

Where t denotes the time for which oscillation is taking place.

Simple Harmonic Motion

Simple harmonic motion (SMH) is defined as a periodic oscillation of an unaltering frequency and a constant amplitude where the magnitude of restoring force on the object is always acting towards its equilibrium position and is directly proportional to its displacement.

Simple harmonic motion is the simplest form of oscillatory motion meaning. It is an ideal condition where the maximum displacement on one side of the equilibrium position is exactly equal to that on the other side. This aspect is also required to define oscillating motion.

SHM is denoted by y, and its mathematical expression is -

y = A sin ωT or A cos ωT

Where A stands for amplitude, ω stands for angular frequency, and T stands for time.

A perfect example of the simple harmonic motion is again the motion of a simple pendulum because, on its displacement in one direction, a proportional restoring force acts on it in the opposite direction.

What are the Types of Oscillation?

Based on the external forces acting on them, simple harmonic motions are classified into three significant categories. Now we will define oscillating motion and its types with a brief description of each.

• Free Oscillation

A free oscillation is an ideal condition where a particle’s motion is not under the influence of any external resistance. It is a motion with a natural frequency of the particle and constant amplitude, energy, and period.

It is an ideal condition because, in reality, every oscillating object undergoes some form of interaction with external conditions resulting in loss of energy. 

An example of free oscillations is the motion of a simple pendulum in a vacuum.

• Damped Oscillation

A damping oscillation is one in which the moving particle gradually loses its kinetic energy on interaction with resistive forces like air or friction. Due to this resistance offered by external forces, the displacement of a particle slowly reduces with time and ultimately reaches its state of rest. 

Damped oscillations are classified based on the difference in energy between the applied restoring force and the restraining force acting on it. Basically, it is an oscillation that fades away with time, which means the oscillations will reduce in their magnitude with time.

The damping can be natural or forced. 

The best example of damped oscillations is a simple pendulum oscillating under natural conditions. Shock absorbers in vehicles are an example of damping devices that reduce the vehicle’s vibrations.

The different types of damped oscillation are: 

Under damped oscillations: Damping constant < 1

Critically damped oscillations: Damping constant = 1

Over damped oscillations: Damping constant > 1

What is Electromagnetic Damping?

Damping plays a crucial role in controlling the object’s motion. Electromagnetic damping uses electromagnetically induced current to control, regulate or slow down the motion of an object without the use of any physical contact. The damping force is basically electromagnetically applied.

The damping technique depends on Eddy current and Electromagnetic induction. A damping force is generated when the Eddy current and magnetic field interact with each other and create a resistive force. And this force opposes the conductor or object’s motion. 

So, the electromagnetic damping force is directly proportional to the following three factors: the induced Eddy Current, the object’s speed and magnetic field’s strength. This means that the faster the object’s motion, the greater the damping. And the slower the object’s motion, the lower the damping, resulting in stopping the object smoothly and effectively. 

The concept of Free Forced Damped Oscillations constitutes a significant portion of Class 11 Physics. It makes up almost 14% of the syllabus with some key concepts essential for competitive exams. 

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