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In damped oscillations, the amplitude is reduced to one-third of its initial value at the end of 100 oscillations. When the oscillator completes 200 oscillations, its amplitude must be _
A. \[\dfrac{{{a_o}}}{2}\]
B. \[\dfrac{{{a_o}}}{4}\]
C. \[\dfrac{{{a_o}}}{6}\]
D. \[\dfrac{{{a_o}}}{9}\]

Last updated date: 15th Jul 2024
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Hint: The amplitude of oscillations steadily decreases over time as a result of radiation from an oscillating mechanism and friction in the system. An oscillator's amplitude (or energy) is reduced by damping, and the oscillations are said to be damped.

Complete answer:
The existence of resistance to motion means that the device is subjected to frictional or damping forces. The damping force works against the wave, performing destructive work on the mechanism and causing energy dissipation. When a body travels through a medium like air, water, or snow, the energy is dissipated by friction and manifests as heat in the body, the ambient medium, or both.
An oscillator may also lose energy by another process. An oscillator's energy can be reduced not only by friction in the device, but also by radiation. The oscillating body produces waves by imparting periodic motion to the particles of the medium in which it oscillates. A tuning fork, for example, generates sound waves in the medium, which reduces the energy.
The amplitude of a damped oscillation continues to decay exponentially.
\[a = {a_o}{e^{ - bt}}\]
where b=damping coefficient.
T=time of one oscillation
A damping coefficient is a property of a material that determines whether it can bounce back or return energy to a device. If the bounce is caused by an unexpected vibration or shock, a material with a high damping coefficient can reduce the reaction. It will absorb the energy and dampen the unfavourable reaction.
Given initially,
\[\dfrac{{{a_0}}}{3} = {a_0}{e^{ - b \times 100T}}\]
\[\dfrac{1}{3} = {e^{ - 100bT}}\]-----(1)
\[a = {a_0}{e^{ - b \times 200T}}\]
\[a = {a_0}{\left[ {{e^{ - 100bT}}} \right]^2}\]
\[a = {a_0} \times {\left[ {\dfrac{1}{3}} \right]^2}\]
From (1)
\[a = \dfrac{{{a_0}}}{9}\]
Hence option D is correct.

Note: Both sounding bodies are subject to dissipative forces; otherwise, the body would lose no energy and, as a result, no sound energy would be emitted. As a result, sound waves are produced by mechanical oscillatory systems emitting radiation. Electromagnetic waves are generated by radiations from oscillating electric and magnetic fields, as we'll see later.