FORCED VIBRATIONS
The vibrations of a body which take place under the influence of an external periodic force acting on it, are called forced vibrations.
The body executing forced vibrations is thus acted upon by three forces: (i) the restoring force, (ii) the frictional (or resistive or damping) force, and (iii) the external periodic force. When an external periodic force is applied on a vibrating body, the body no longer vibrates with its own natural frequency, but it gradually acquires the frequency of the applied periodic force. The external applied force is also called the driving force.
The amplitude of the forced vibrations depends on the frequency of the external force. If the frequency of the external force is different from the natural frequency of the body, the amplitude of oscillations is very small which die out soon due to the presence of damping forces. But if the frequency of the external force is exactly equal to the natural frequency of the vibrating body, the amplitude of oscillations is very large. In absence of damping forces, the amplitude of vibrations at resonance is infinite.
Note: The amplitude of forced vibrations of a body does not remain constant due to the presence of damping forces of the surrounding medium. However, it is possible to keep the amplitude of vibrations constant by applying an external periodic force such that the external periodic force compensates for the loss of energy in each vibration due to the damping forces. The vibrations of the body then are called sympthatic vibrations.
Examples of forced vibrations
(1) When the stem of a vibrating tuning fork is pressed against the top of a table, the tuning fork produces forced vibrations in the table top. The table top has a much larger vibrating area than the tuning fork, so the forced vibrations of the table top send forth a greater energy and produces a louder (or more intense) sound than that produced by the tuning fork.
(2) The vibrations produced in the diaphragm of a microphone sound box with frequencies corresponding to the speech of the speaker, are forced vibrations.
(3) In a guitar, when an artist plays on its strings by his fingers, the vibrations in them cause forced vibrations in the air enclosed in its hollow box.
(4) In stringed instruments provided with a hollow sound box containing air, strings pare made to vibrate by plucking which La produce forced vibrations in the air of the sound box. The surface area of the air in the sound box is large, so the forced vibrations of air send forth a greater energy and cause a loud sound.
RESONANCE (a special case of forced vibrations)
When a body vibrates in a medium under the influence of an external periodic force of frequency exactly equal to the natural frequency of vibration of the body, it is said to execute resonant vibrations. These vibrations are of large amplitude. The amplitude at resonance depends or the frictional forces. Thus,
Resonance is a special case of forced vibrations. When the frequency of the externally applied periodic force on a body is equal to its natural frequency, the body readily begins to vibrate with an increased amplitude. This phenomenon is known as resonance. The vibrations of large amplitude are called resonant vibrations.
Suppose a body is set to vibrate freely and it vibrates with a frequency f. It is the natural frequency of vibration of the body. Now let a periodic force of frequency n be applied on the vibrating body. If n = f, resonance takes place and the body starts getting energy from the source of the applied periodic force, so its amplitude of vibration is increased. In case n is greater or less than f. (i.e.. not equal to f), there will be no resonance and the body will execute only forced vibrations of small amplitude.
Condition for resonance
Resonance occurs only when the applied force causes forced vibration in the body and the frequency of the applied force is exactly equal to the natural frequency of the vibrating body.
At resonance, a loud sound is heard because the body vibrating with a very large amplitude sends forth a large amount of energy in the medium.
SOME EXAMPLES OF RESONANCE
(1) Resonant vibrations of pendulums: As discussed earlier in experiment (2), if two pendulums of the same length are suspended from a rubber string and one pendulum is made to vibrate, the other pendulum also starts vibrating with a large amplitude and in same phase because of resonance.
(2) Resonance in machine parts: When a vehicle is driven, the piston of the engine makes to and fro motion at a frequency depending upon the speed of the vehicle. The vibrations caused by the movement of the piston are communicated to all parts of the vehicle. It is just possible that at a certain speed of the vehicle some parts (or frame) of the vehicle may have a natural frequency of vibration equal to the frequency of the to and fro movement of the piston. At this particular speed of the vehicle, that particular part or frame starts vibrating vigorously due to resonance and a rattling sound is heard. If this part is not tightly fixed, such vibrations may cause the part to drop out. To stop these vibrations, the speed of the vehicle is changed, so that the condition of resonance is disturbed.
Note: Thus, resonance can occur in all kinds of machines while operating in a particular condition.
(3) Resonance in a stretched string and sound box of musical instruments and sonometer: A vibrating string by itself produces a very weak sound which cannot be heard at a distance. Therefore, all musical stringed instruments such as guitar and sonometer are provided with a sound box (i.e. a hollow chamber). The box is so constructed that the column of air inside it. has a natural frequency which is same as that of the string stretched on it, so that when the string is made to vibrate, the air column inside the box is set into forced vibrations, the frequency of which is the same as that of the string. Since the sound box has a large area, it sets a large volume of air into vibrations, so due to resonance, a loud sound n is produced.
(4) Resonance in air column and tuning fork: As described in experiment (3), when the frequency of the air column becomes equal to the frequency of the tuning fork vibrating over its mouth, a loud sound is heard due to resonance.
(5) Resonance in a bridge: When a troop crosses a suspension bridge, the soldiers are asked to break their steps. The reason is that when soldiers march in steps, each soldier exerts a periodic force in the same phase and therefore the bridge executes the forced vibrations of frequency equal to the frequency of their steps. Now if the frequency of the steps becomes equal to the natural frequency of the bridge, the bridge will vibrate with a large amplitude due to resonance and the suspension bridge may collapse. (6) Resonance in radio and TV receivers: The tuning of a radio and TV receivers is based button resonance. The radio and TV receivers have electronic circuits which produce electromagnetic vibrations, the frequency of which can be changed by varying the values of the electronic components (either the capacity of condenser or the inductance of coil)*. When we want to tune a radio (or TV) receiver, we merely adjust the values of the electronic components to produce vibrations of frequency equal to that of the radio waves which we want to receive. When both the frequencies match, resonance occurs and only the energy of the signal of that particular art frequency is received from the waves present in space, leaving the signals of other frequencies which do not match with the frequency of the receiver circuit. The signal received is then amplified in the receiver set.* Usually the capacity of condenser is changed.
Demonstration of resonance
The phenomenon of resonance can be demonstrated by the following experiments:
Experiment (1)-Resonance with tuning forks: Mount two identical tuning forks A and B of the same frequency on two separate sound boxes such that their open ends face each other as shown in Fig. 7.8. When the prong of one of the tuning forks say, A is struck on a rubber pad, it starts vibrating. On putting the tuning fork A on its sound box, we find that the other tuning fork B also starts vibrating and a loud sound is heard.
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Resonance with tuning forks |
Reason - The vibrating tuning fork A produces forced vibrations in the air column of its soundbox. These vibrations are of large amplitude because of large surface area of air in the soundbox. They are communicated to the sound box of the fork B. The air column of B starts vibrating with the frequency of fork A. Since the frequency of these vibrations is the same as the natural frequency of the fork B, the fork B picks up these vibrations and starts vibrating under resonance. Thus the two sound boxes help in communicating the vibrations and in increasing the amplitude of vibrations.
Difference between Forced vibrations and Resonant vibrations
Forced vibrations1. The vibrations of a body under an external periodic force of frequency different from the natural frequency of the body, are called forced vibrations.
2. The amplitude of vibration is small.
3. The vibrations of the body are not in phase with the external periodic force.
4. These vibrations last for a very small time after the periodic force has ceased to act.
Resonant vibrations
1. The vibrations of a body under an external periodic force of frequency exactly equal to the natural frequency of the body, are called resonant vibrations.
2. The amplitude of vibration is very large.
3. The vibrations of the body are in phase with the external periodic force.
4. These vibrations last for a long time after the periodic force has ceased to act.
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