Notes On Free, Damped and Forced Vibrations - ICSE Class 10 Physics
 If a tuning fork is struck with a rubber hammer, it vibrates with a certain frequency. This frequency does not change even when the tuning fork is struck with the same rubber hammer, harder than earlier. The oscillations of a body, when set into vibration and left to itself, are called free vibrations or natural vibrations. The frequency of a body executing natural vibrations is called its natural frequency. Every vibrating body has its own natural frequency. The natural frequency of a tuning fork depends on its dimensions. The natural frequency of a simple pendulum depends on its length for a given place. The time period of oscillation of a body is called its natural time period. Once set into oscillation, a body executes vibrations with constant amplitude when there are no dissipative forces. Here, amplitude is the maximum displacement of the body from its mean position. When a simple pendulum is slightly disturbed and released, it oscillates. The frequency of oscillation, ‘n’of a simple pendulum is directly proportional to the square root of acceleration due to gravity ‘g’ at a place, and inversely proportional to the square root of length ‘L’ of the pendulum. For a simple pendulum of length one metre, oscillating at a place having acceleration due to gravity of [9.8 metre per second square], the frequency of the pendulum is [0.5] hertz. Consider a spring of negligible mass, with one end fixed rigidly to a support, and a load hung at the other end. When the load is pulled down slightly and released, it executes oscillations vertically. The natural frequency, ‘n,’ of the spring depends on two factors, its stiffness constant ‘k,’ and mass ‘m’ of the load attached to the spring. The stiffness constant can be different for different springs. Therefore, for the same load attached to them, they oscillate with different natural frequencies and time periods. When the keys of a piano are struck, various strings of different natural frequencies are set into vibrations. Consider a closed organ pipe, which is a pipe closed at one end and open at the other. The air column in it can vibrate with a certain natural frequency ‘n,’ which is inversely proportional to length ‘L’ of the air column. The air column in it can vibrate in different modes of vibration with frequencies in the ratio of 1:3:5... and so on. Consider an open organ pipe, which is a pipe open at both the ends. The air column in it can vibrate with a certain frequency ‘n,’ which is inversely proportional to length ‘L’ of the air column. The air column in it can vibrate in different modes of vibration with frequencies in the ratio of 1:2:3 ... and so on. When a string is held under tension between two rigid supports and plucked in the middle, it executes transverse vibrations. Stationary transverse waves are set up in the string. In the least frequency mode, the string vibrates with one loop with a certain frequency ‘n’. This is the least frequency of vibration that can be set up in the string, and is called its fundamental frequency, ‘n’. The fundamental frequency, ‘n,’ of the string is inversely proportional to its length ‘L” and radius “R” of its cross section, directly proportional to the square root of the tension ‘T’ in it, and inversely proportional to the square root of the density, ‘d,’ of its material. When the string is plucked at one-fourth of its length from one end, it vibrates with two loops, and its frequency is equal to “2n,” which is called the first overtone. When the string is plucked at one-sixth of its length from one end, it vibrates with three loops, and here, its frequency is equal to “3n”, which is called the third overtone. Therefore, we see that as the number of loops in the string increases, its frequency of vibration increases. In stringed instruments like the guitar, we see that the string, held under some tension between two rigid supports, executes transverse vibrations when excited. Now let us understand the nature of free vibrations. When a simple pendulum is pulled slightly aside and left to itself, it should vibrate with a constant amplitude and frequency forever. Such vibrations can be realised only in vacuum. However, the simple pendulum executes oscillations with decreasing amplitude because of the effect of the surrounding medium. The surrounding medium offers resistance to the motion, and hence, the vibrating body loses energy continuously to the surroundings. Therefore, the amplitude of the body continuously decreases. The periodic vibrations of a body decreasing in amplitude are called damped vibrations. The rate at which energy is lost by the body depends on the surrounding medium, as well as on the size and shape of the vibrating body. Let us now learn about forced vibrations of a body. Consider three pendulums A, B and C of different lengths suspended from a common support. A, B and C have different natural frequencies of oscillation as their lengths differ. Now set A into oscillation. You will see that after some time, B and C also start oscillating. However, B and C do not oscillate at their natural frequencies. They oscillate with a frequency equal to that of A. Thus, B and C oscillate under the influence of A. The oscillations of B and C are called forced vibrations. When a body vibrates under the influence of an external periodic force with a frequency equal to that of the external periodic force, then the vibrations of the body are called forced vibrations. Let us now look at some other examples of forced vibrations. When a tuning fork is excited and pressed against a table top, the table top vibrates. Here, the tuning fork is the external periodic force, and the table top vibrates under its influence. The vibrations executed by the table top are forced vibrations. The table top vibrates with a frequency equal to the external periodic force, that is, the frequency of the tuning fork. The table vibrates with decreasing amplitude. The vibrations of the table top are spread over a large area. Hence, the vibrating table top comes into contact with a greater number of air particles than the vibrating tuning fork. As a result, a louder sound is produced by the vibrating table top than by the vibrating tuning fork. Thus, in forced vibrations, when a body vibrates under the influence of an external periodic force, it does not vibrate with its natural frequency, but with the natural frequency of the external periodic force.  Stringed musical instruments like the guitar have a hollow sound box that contains air. When the string of a guitar is excited and it vibrates, the air in the hollow sound box vibrates under the influence of the vibrating string. The air vibrates with a frequency equal to the natural frequency of the vibrating string, which is the external periodic force, and produces a loud musical note. The frequencies produced in the diaphragm of a gramophone sound box that correspond to the frequencies of the tones conveyed by the gramophone record, are forced vibrations.

#### Summary

 If a tuning fork is struck with a rubber hammer, it vibrates with a certain frequency. This frequency does not change even when the tuning fork is struck with the same rubber hammer, harder than earlier. The oscillations of a body, when set into vibration and left to itself, are called free vibrations or natural vibrations. The frequency of a body executing natural vibrations is called its natural frequency. Every vibrating body has its own natural frequency. The natural frequency of a tuning fork depends on its dimensions. The natural frequency of a simple pendulum depends on its length for a given place. The time period of oscillation of a body is called its natural time period. Once set into oscillation, a body executes vibrations with constant amplitude when there are no dissipative forces. Here, amplitude is the maximum displacement of the body from its mean position. When a simple pendulum is slightly disturbed and released, it oscillates. The frequency of oscillation, ‘n’of a simple pendulum is directly proportional to the square root of acceleration due to gravity ‘g’ at a place, and inversely proportional to the square root of length ‘L’ of the pendulum. For a simple pendulum of length one metre, oscillating at a place having acceleration due to gravity of [9.8 metre per second square], the frequency of the pendulum is [0.5] hertz. Consider a spring of negligible mass, with one end fixed rigidly to a support, and a load hung at the other end. When the load is pulled down slightly and released, it executes oscillations vertically. The natural frequency, ‘n,’ of the spring depends on two factors, its stiffness constant ‘k,’ and mass ‘m’ of the load attached to the spring. The stiffness constant can be different for different springs. Therefore, for the same load attached to them, they oscillate with different natural frequencies and time periods. When the keys of a piano are struck, various strings of different natural frequencies are set into vibrations. Consider a closed organ pipe, which is a pipe closed at one end and open at the other. The air column in it can vibrate with a certain natural frequency ‘n,’ which is inversely proportional to length ‘L’ of the air column. The air column in it can vibrate in different modes of vibration with frequencies in the ratio of 1:3:5... and so on. Consider an open organ pipe, which is a pipe open at both the ends. The air column in it can vibrate with a certain frequency ‘n,’ which is inversely proportional to length ‘L’ of the air column. The air column in it can vibrate in different modes of vibration with frequencies in the ratio of 1:2:3 ... and so on. When a string is held under tension between two rigid supports and plucked in the middle, it executes transverse vibrations. Stationary transverse waves are set up in the string. In the least frequency mode, the string vibrates with one loop with a certain frequency ‘n’. This is the least frequency of vibration that can be set up in the string, and is called its fundamental frequency, ‘n’. The fundamental frequency, ‘n,’ of the string is inversely proportional to its length ‘L” and radius “R” of its cross section, directly proportional to the square root of the tension ‘T’ in it, and inversely proportional to the square root of the density, ‘d,’ of its material. When the string is plucked at one-fourth of its length from one end, it vibrates with two loops, and its frequency is equal to “2n,” which is called the first overtone. When the string is plucked at one-sixth of its length from one end, it vibrates with three loops, and here, its frequency is equal to “3n”, which is called the third overtone. Therefore, we see that as the number of loops in the string increases, its frequency of vibration increases. In stringed instruments like the guitar, we see that the string, held under some tension between two rigid supports, executes transverse vibrations when excited. Now let us understand the nature of free vibrations. When a simple pendulum is pulled slightly aside and left to itself, it should vibrate with a constant amplitude and frequency forever. Such vibrations can be realised only in vacuum. However, the simple pendulum executes oscillations with decreasing amplitude because of the effect of the surrounding medium. The surrounding medium offers resistance to the motion, and hence, the vibrating body loses energy continuously to the surroundings. Therefore, the amplitude of the body continuously decreases. The periodic vibrations of a body decreasing in amplitude are called damped vibrations. The rate at which energy is lost by the body depends on the surrounding medium, as well as on the size and shape of the vibrating body. Let us now learn about forced vibrations of a body. Consider three pendulums A, B and C of different lengths suspended from a common support. A, B and C have different natural frequencies of oscillation as their lengths differ. Now set A into oscillation. You will see that after some time, B and C also start oscillating. However, B and C do not oscillate at their natural frequencies. They oscillate with a frequency equal to that of A. Thus, B and C oscillate under the influence of A. The oscillations of B and C are called forced vibrations. When a body vibrates under the influence of an external periodic force with a frequency equal to that of the external periodic force, then the vibrations of the body are called forced vibrations. Let us now look at some other examples of forced vibrations. When a tuning fork is excited and pressed against a table top, the table top vibrates. Here, the tuning fork is the external periodic force, and the table top vibrates under its influence. The vibrations executed by the table top are forced vibrations. The table top vibrates with a frequency equal to the external periodic force, that is, the frequency of the tuning fork. The table vibrates with decreasing amplitude. The vibrations of the table top are spread over a large area. Hence, the vibrating table top comes into contact with a greater number of air particles than the vibrating tuning fork. As a result, a louder sound is produced by the vibrating table top than by the vibrating tuning fork. Thus, in forced vibrations, when a body vibrates under the influence of an external periodic force, it does not vibrate with its natural frequency, but with the natural frequency of the external periodic force.  Stringed musical instruments like the guitar have a hollow sound box that contains air. When the string of a guitar is excited and it vibrates, the air in the hollow sound box vibrates under the influence of the vibrating string. The air vibrates with a frequency equal to the natural frequency of the vibrating string, which is the external periodic force, and produces a loud musical note. The frequencies produced in the diaphragm of a gramophone sound box that correspond to the frequencies of the tones conveyed by the gramophone record, are forced vibrations.

#### Activities

 Activity 1 Acoustics.salford has created an animation to understand the different types of vibrations i.e., Free, Damped and Forced Vibrations. Go to Activity

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