Question

In damped oscillations, the amplitude after 50 oscillations is 0.8 ${a_0}$ ​, where${a_0}$ is the initial amplitude, then the amplitude after 150 oscillations is

Answer 1

Hint: In this question, we will use the relation between the initial amplitude, time period and the amplitude. By substituting the given equation we will get the required result. Further, we will discuss the basics of a wave and superposition of wave, for our better understanding.

Formula used:
$a = {a_0}\exp ( - \alpha t)$

Complete step by step solution:
As we know that, the damped simple harmonic motion, the energy of the oscillator dissipates continuously.
 As we know, the amplitude a, is given by:
$a = {a_0}\exp ( - \alpha t)$
Now, by substituting the given values in the above equation:
${a_{50}} = {a_0}\exp ( - \alpha \times 50T) = 0.80a{}_0$
Here, T is the period of oscillation.
 ${a_{150}} = {a_0}\exp ( - \alpha \times 150T) = a{}_0{\left( {0.8} \right)^3}$
$\therefore {a_{150}} = 0.512{a_0}$
Therefore, we get the required result

Additional Information: Waves involve the transfer of energy without the transfer of the matter. So, it can be said that waves can be described as a disturbance that travels through a medium, transporting energy from one location to another location without transfer of matter.
Further, the frequency is defined as the number of waves that pass a fixed point in unit time. It can also be defined as the number of cycles or vibrations undergone during one unit of time.
Two waves are said to be coherent if they are moving with the same frequency and have constant phase difference.
The summation or adding or subtraction of all the waves travelling in a particular medium, gives us the superposition of waves. If the direction or amplitude of the waves are opposite then the superposition of waves is calculated by subtracting the waves, whereas if the two waves are travelling in the same direction or have the same amplitude the resultant is given by adding up the two or more waves.
The S.I unit of frequency is Hertz or Hz and the unit of wavelength is meter or m. Furthermore we also know the S.I unit of time which is given by second or s.
Phase of a wave specifies the location of a point within a wave cycle of a repetitive waveform. Generally, the phase differences between two or more sound waves are important. When two sound waves combine, like- the difference between the phases of the two waves is important in determining the resulting waveform.

Note: The phase of the wave can be positive or negative depending on its direction of propagation. A sine wave starts from zero, whereas the cosine wave starts from one. A wave which has the same amplitude but opposite orientation will cancel out each other and thereby give zero output.