Answer

Verified

392.4k+ views

**Hint:**Start by assuming the time, when the particle will be at the mean position .Then using the SHM equation for displacement of particle $x=A\sin(\omega t)$, find the time for the particle to complete $\dfrac{5}{8}$ oscillations

**Formula:**$x=A\sin(\omega t)$

**Complete answer:**

SHM or simple harmonic motion is the motion caused by the restoring force; it is directly proportional to the displacement of the object from its mean position. Clearly, this is a reactive force, which is always directed towards the mean. Let us assume the equation for displacement of particle to be given as $x=A\sin(\omega t)$, then the acceleration of the particle is given by, $a(t)=-\omega^{2}x(t)$. We know that $\omega$ is the angular velocity and is given as $\omega=\dfrac{2\pi}{T}$, where $T$ is the time period of the motion.

Given that the time period of the particle is $T$. Let the total distance covered by the particle during the time period T be $4\;A$. Then the distance covered during $\dfrac{5}{8}$ can be written as $\dfrac{5}{8}\times 4A=\dfrac{5A}{2}$

Also, $\dfrac{5A}{2}$ can be written in terms of $\dfrac{A}{2}$ as $\dfrac{5A}{2}=2A+\dfrac{A}{2}$

Now, from the equation of the particle, we can find the time $t$ taken due to $\dfrac{A}{2}$ distance. Consider the equation$\dfrac{A}{2}=A sin\omega t$

$\implies \dfrac{A}{2}=A sin \dfrac{2\pi}{T}t$

$\implies \dfrac{1}{2}=sin\dfrac{2\pi}{T}t$

$\implies sin\dfrac{\pi}{6}=sin\dfrac{2\pi}{T}t$

$\implies t=\dfrac{T}{12}$

Similarly, the time $t\prime$ taken to cover $2\;A$ is given as, $2A=Asin\dfrac{2\pi}{T}t\prime$

$\implies 2\times sin \dfrac{\pi}{2}=sin\dfrac{2\pi}{T}t\prime$

$\implies t\prime=\dfrac{T}{2}$

Then total time take to complete $\dfrac{5}{8}$ oscillations is given as $T\prime=t+t\prime$

$\implies T\prime=\dfrac{T}{12}+\dfrac{T}{2}$

$\therefore T\prime=\dfrac{7T}{12}$

**Thus the correct answer is option \[D.\dfrac{7T}{12}\]**

**Note:**

Remember SHM motions are sinusoidal in nature. Here, we are assuming, the particle is at mean when, $t=0$. This makes the further steps easier. Then, it will take time $T$ for the particle to cover one oscillation. For simplification, we are finding the time taken due to the small parts of the oscillation

Recently Updated Pages

10 Examples of Friction in Our Daily Life

10 Examples of Evaporation in Daily Life with Explanations

The marks obtained by 50 students of class 10 out of class 11 maths CBSE

Explain the law of constant proportion in a simple way

The magnetic field due to a short magnet at a point class 12 physics JEE_Main

A transparent thin plate of polaroid is placed on another class 12 physics JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Mention the different categories of ministers in the class 10 social science CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Who is the executive head of the Municipal Corporation class 6 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which monarch called himself as the second Alexander class 10 social science CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Write an application to the principal requesting five class 10 english CBSE