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**Hint:**Amplitude of a damped oscillator decreases exponentially. For this first calculate the amplitude of the damped oscillator when it decreases o.9 times at 5 sec. Similarly find the decrease in amplitude at 10s using the concept amplitude decreases exponentially. By comparing both the equations we will get the value of $\alpha $ .

**Formula used:**

$A={{A}_{0}}{{e}^{-kt}}$

where, A is the new amplitude.

${A_0}$ is the initial amplitude.

t is the time taken to decrease the amplitude.

k is the damping constant.

**Complete answer:**

The amplitude of oscillations gradually decreases to zero as a result of frictional forces, arising due to the viscosity of the medium in which the oscillator is moving. The motion of the oscillator is damped by friction and therefore is called a damped harmonic oscillator. .

If damping is taken into account then a harmonic oscillator experiences a restoring force and a damping force proportional to the velocity.

Then the equation of motion of the damped harmonic oscillator is given by,

$\dfrac{{{d}^{2}}x}{d{{t}^{2}}}=-\gamma \dfrac{dx}{dt}-Cx$

$\dfrac{{{d}^{2}}x}{d{{t}^{2}}}+\gamma \dfrac{dx}{dt}+Cx=0$

$\dfrac{{{d}^{2}}x}{d{{t}^{2}}}+2k\dfrac{dx}{dt}+{{\omega }_{0}}x=0$

where, $2k=\dfrac{\gamma }{m}$ and ${{\omega }_{0}}=\sqrt{\dfrac{C}{m}}$

Then amplitude A is,

$A={{A}_{0}}{{e}^{-kt}}$

substituting the values of amplitude, initial amplitude and time in the above equation we get,

0.9${A_0}$=${{A}_{0}}{{e}^{-5k}}$

$\Rightarrow $ 0.9=${{e}^{-5k}}$

Taking natural log on both sides,

$\Rightarrow $ ln(0.9)= -5k

At time t=15s equation becomes,

$A={{A}_{0}}{{e}^{-15k}}={{A}_{0}}{{e}^{3ln(0.9)}}$

= ${{A}_{0}}{{(0.9)}^{3}}=0.729{{A}_{_{0}}}$

**Hence,option (B) is correct.**

**Additional information:**

The frictional force, acting on a body opposite to the direction of its motion, is called damping force. Such a force reduces the velocity and the kinetic energy of the moving body and so it is also called retarding or dissipative force. Damped oscillations means a decrease in intensity of oscillations with time. The force of friction retards the motion so that the system does not oscillate indefinitely. The friction reduces the mechanical energy of the system, the motion is said to be damped and this damping reduces amplitude of the vibratory motion.

**Note:**

The damping lowers the natural frequency of the object and also reduces the magnitude of amplitude of the wave. Amplitude of a damped oscillator decreases exponentially That is, the thermal energy of the damping force is removed from the system Thus damped oscillations means a decrease in intensity of oscillations with time.

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