6.6
Free and Forced Vibrations
Superposition of Waves · Maharashtra HSC
Theory
Natural Frequency
The frequency at which an object tends to vibrate when hit, plucked or somehow disturbed is its natural frequency. In such vibrations the object is not under the influence of any outside force.
Free Vibrations
In free vibration the body is first given an initial displacement and the force is then withdrawn. The body starts vibrating and continues the motion on its own; no external force acts on it further to keep it in motion. Free vibration of a system means that the system vibrates at its natural frequency.
Examples: a simple pendulum pulled aside and released; a guitar string plucked at some point.
In free vibrations the body continuously loses energy against the frictional resistance of the surrounding medium, so the amplitude goes on decreasing, the vibrations eventually stop and the body comes to rest.
Forced Vibrations
In forced vibrations an external periodic force is applied to a body whose natural period is different from the period of the force. The body is made to vibrate with a frequency equal to that of the externally impressed force.
Examples: the vibrations in a drill machine and in a washing machine; the vibrations produced in a table top when a vibrating tuning fork is held against it. Two tuning forks of different frequencies held on the same table top produce different vibrations in it, because these are forced vibrations at the frequency of whichever fork is driving them.
The amplitude of forced vibrations depends upon the difference between the frequency of the external periodic force and the natural frequency of the body. If this difference is small the amplitude is large, and if it is large the amplitude is small.
Resonance
If the frequency of the external periodic force exactly matches the natural frequency of the body, the amplitude of vibration is maximum. This is resonance.
An object vibrating with its natural frequency can set a nearby object vibrating: the second object absorbs the energy transmitted by the first and starts vibrating if the natural frequencies of the two match. The second object is then said to undergo forced vibrations. Strings and air columns also undergo forced oscillations when the frequency of an external source of sound is close to their natural frequency; resonance occurs and a louder sound is heard.
Distinguishing Between Free and Forced Vibrations
| Free vibrations | Forced vibrations |
|---|---|
| Produced when a body is disturbed from its equilibrium position and released | Produced by an external periodic force of any frequency |
| To start them, the force is required initially only | A continuous external periodic force is required; if it is stopped, the forced vibrations also stop |
| The frequency depends on the natural frequency of the body | The frequency depends on the frequency of the external periodic force |
| Energy of the body remains constant in the absence of friction and air resistance; damping forces make the total energy decrease | Energy of the body is maintained constant by the external periodic force |
| Amplitude of vibrations decreases with time | Amplitude is small but remains constant as long as the external periodic force acts |
| Vibrations stop sooner or later depending on the damping force | Vibrations stop as soon as the external periodic force is stopped |
Where the question asks for two or four points, take them from the top of this table in order.
Standard Results
- Free vibration: the body oscillates at its own natural frequency, with decaying amplitude.
- Forced vibration: the body oscillates at the frequency of the driving force, with constant amplitude while the force acts.
- Resonance: driving frequency equal to natural frequency, amplitude maximum, sound loudest.
Theory Questions Asked on This Subtopic
Real board questions. Answers are at the end of the page.
- 1
Distinguish between free vibrations and forced vibrations.
Maharashtra HSC board question — asked in March 2013Answer 1 ↓ - 2
Distinguish between free vibrations and forced vibrations (Two points).
Maharashtra HSC board question — asked in February 2023Answer 2 ↓
Answers & Solutions
Work the questions above first, then check yourself here. Each answer is set out the way it should be written in the paper.
Sr. No. Free vibrations Forced vibrations i. Free vibrations are produced when a body is disturbed from its equilibrium position and released. Forced vibrations are produced by an external periodic force of any frequency. ii. To start free vibrations, the force is required initially only. Continuous external periodic force is required. If external periodic force is stopped, then forced vibrations also stop. iii. The frequency of free vibrations depends on the natural frequency. The frequency of forced vibrations depends on the frequency of the external periodic force. iv. Energy of the body remains constant in the absence of friction, air resistance, etc. Due to damping forces, total energy decreases. Energy of the body is maintained constant by the external periodic force. v. Amplitude of vibrations decreases with time. Amplitude is small but remains constant as long as external periodic force acts on it. vi. Vibrations stop sooner or later depending on the damping force. Vibrations stop as soon as external periodic force is stopped. [Any four differences]
[From 6.6, Q1 (Any Two Points)]
Sr. No. Free vibrations Forced vibrations i. Free vibrations are produced when a body is disturbed from its equilibrium position and released. Forced vibrations are produced by an external periodic force of any frequency. ii. To start free vibrations, the force is required initially only. Continuous external periodic force is required. If external periodic force is stopped, then forced vibrations also stop. iii. The frequency of free vibrations depends on the natural frequency. The frequency of forced vibrations depends on the frequency of the external periodic force. iv. Energy of the body remains constant in the absence of friction, air resistance, etc. Due to damping forces, total energy decreases. Energy of the body is maintained constant by the external periodic force. v. Amplitude of vibrations decreases with time. Amplitude is small but remains constant as long as external periodic force acts on it. vi. Vibrations stop sooner or later depending on the damping force. Vibrations stop as soon as external periodic force is stopped. [Any four differences]