Oscillations: Types and Equations

Oscillation is the repeated or periodic motion of a body about a mean position.

• It is a time-dependent measure of some recurrent variation.
• It can be calculated in terms of a condition of equilibrium.
• Oscillation can be observed when the body moves in a to and fro motion around the same point at a uniform interval of time.
• This type of motion is also known as oscillatory motion.
• The point about which the body moves is called the mean or equilibrium position.
• Mechanical oscillations are also called vibrations and similarly, oscillatory motion is also called vibratory motion.

Some examples of oscillatory motion are 

• The motion of the bob of a pendulum
• The motion of the piston of an automobile engine 
• The motion of the mass attached to a spring, etc.

Every oscillatory motion is a periodic motion, but every periodic motion is not oscillatory.

Key Terms: Oscillation, oscillatory motion, equilibrium position, constant motion, equilibrium, damped oscillations, forced oscillations, free oscillations, kinetic energy, periodic motion, vibratory motion

---

Oscillations Calculation

[Click Here for Sample Questions]

If a body performing the oscillatory motion, then the following terms can be calculated:

• Time Period: The time taken by an oscillating body to complete one oscillation is called its time period or period of oscillations. It is denoted by T and its SI unit is second.
• Frequency: The number of oscillations completed by an oscillating body is called the frequency of the body. It is denoted by η or f.

The relation between time period and frequency is given by

frequency (η) = 1/ time period (T)

A block of mass m attached to a spring of spring constant k performs oscillatory motion and the time period is given by

\(T=2\pi \sqrt {2m\over k}\)

The motion of the bob of the simple pendulum is also an example of oscillatory motion.

Let l be the length of the string of the pendulum, then its time period is given by

\(T=2\pi \sqrt {l\over g}\)

The time period of the oscillation of a torsional pendulum having a moment of inertia I of the body and torsional constant c is given by

\(T=2\pi \sqrt {I\over c}\)

The time period of the oscillation of liquid in a U-tube of length l, is given by

\(T=2\pi \sqrt {l\over 2g}\)

The video below explains this:

Simple Harmonic Motion Detailed Video Explanation:

Also Read:

---

Simple Harmonic Motion

[Click Here for Previous Year's Questions]

Simple Harmonic Motion is one of the most simple forms of oscillatory motion that occurs frequently in nature.

A particle is said to execute simple harmonic motion (SHM), if it moves to and fro about a fixed point or mean position under the action of restoring force (F) which is directly proportional to its displacement (x) from the mean position, and is always directed towards the mean position.

It is given by

F = -kx

where

• k is the force constant or restoring constant and the negative sign indicates that force resists growth in x.
• N/m is the SI unit of force constant.

This force is known as the restoring force, and it pulls the particle back to its equilibrium position as opposing displacement increases.

---

Oscillations Types

[Click Here for Sample Questions]

Three of the most important types of simple harmonic motion are:

Free Oscillation

A system is said to execute free oscillations if on being disturbed from its mean position, it oscillates itself with natural frequency.

• When the damping in the system is equal to zero then the amplitude remains fixed.
• However, this particular theory is applicable only in cases where damping always occurs.
• In order to exclude the interference of outside forces the deduction in forces can be referred to as damping as a result of which energy, amplitude, and frequency all three of them remain fixed.

Damped Oscillations

The oscillation in which the amplitude decreases gradually with time is called Damped Oscillation.

• The decrease in amplitude of oscillations is due to air drag and friction at the support, which opposes the motion of the pendulum.
• Damping force is a frictional force that acts on a body in the opposite direction of its motion.
• The Velocity and Kinetic Energy of a moving body is reduced by the damping force.
• Damping or dissipative forces are non-conservative in nature and are caused by the viscosity or friction in the medium.
• When body velocities are low, the damping force is found to be proportional to particle velocity.

Different types of damped oscillations are:

• Under damped oscillations
• Critically damped oscillations
• Over damped oscillations

Forced Harmonic Oscillator

When a damped harmonic oscillator is subjected to an additional periodic force, the oscillating system is referred to as a driven or forced harmonic oscillator.

• Its oscillations are referred to as forced oscillations or driven oscillations.
• For example, when a person is on a swing, he/she has to constantly push the swing to keep it moving at a constant pace.
• Otherwise, the speed will start to decrease.

---

Oscillations Resonance

[Click Here for Previous Year's Questions]

Resonance is the phenomenon of an increase in amplitude when the driving force is close to the oscillator's native frequency.

• When the frequency of applied force equals the natural frequency of the oscillator without damping, resonance occurs.
• It is a phenomenon in which one external vibrating system produces a higher amplitude oscillation of another system at a specific frequency.
• A resonant frequency is referred to as the frequency at that particular resonance level.
• The resonance frequency can be detected, for example, when tuning a guitar with the help of another guitar.
• The amplitude of the string's vibration is largest in this situation.
• The vibrational energy stored in the system is the cause of significant amplitude oscillations created at those resonant frequencies.

There are various types of resonance which are as follows:

• Mechanical Resonance
• Acoustic Resonance
• Orbital Resonance
• Electrical Resonance
• Particle Resonance
• Optical Resonance

Also Read: