Answer
Verified
382.5k+ views
Hint: First of all we will see the equation of displacement of a particle executing Simple Harmonic Motion. Then, on differentiating that equation with respect to time, we get the equation of velocity. On comparing the phase constant for both the terms, that is, displacement and velocity we will be able to determine the phase difference between them at any given time instant.
Complete step-by-step answer:
Let the displacement of the particle from the point about which it is executing simple harmonic motion be given by the term $X$ .
Now, if the particle has an initial phase of $\theta $ and angular velocity $\omega $, then the displacement equation of the particle could be written as follows:
$\Rightarrow X=A\sin (\omega t+\theta )$ [Let this expression be equation number (1)]
Where, A is the maximum displacement about the point of execution of simple harmonic motion. This maximum displacement is termed as Amplitude.
Now, we know the differential of displacement of a particle with respect to time gives us the particle’s velocity. Thus, on differentiating equation number (1) with respect to time, we get:
$\begin{align}
& \Rightarrow \dfrac{dX}{dt}=\dfrac{d[A\sin (\omega t+\theta )]}{dt} \\
& \Rightarrow v=A\omega \cos (\omega t+\theta ) \\
\end{align}$
Here, we can write the term obtained for velocity as:
$\Rightarrow A\omega \cos (\omega t+\theta )=-A\omega \sin (\omega t+\theta +\dfrac{\pi }{2})$ [Using: $\cos \left( \dfrac{\pi }{2}+x \right)=-\sin x$ ]
$\therefore v=-A\omega \sin (\omega t+\theta +\dfrac{\pi }{2})$ [Let this expression be equation number (2)]
On comparing equation number (1) and (2), we get:
The difference in the phase of velocity and displacement as $\dfrac{\pi }{2}$ .
Therefore, for a particle executing Simple Harmonic Motion, the phase difference between velocity and displacement is equal to $\dfrac{\pi }{2}$ radian.
So, the correct answer is “Option A”.
Note: We see that the difference in phase of velocity and displacement is independent of any parameter. That is, at any given instant this difference remains the same. If we differentiate the velocity equation again, we will see that the phase difference between displacement and acceleration is $\pi $ radian.
Complete step-by-step answer:
Let the displacement of the particle from the point about which it is executing simple harmonic motion be given by the term $X$ .
Now, if the particle has an initial phase of $\theta $ and angular velocity $\omega $, then the displacement equation of the particle could be written as follows:
$\Rightarrow X=A\sin (\omega t+\theta )$ [Let this expression be equation number (1)]
Where, A is the maximum displacement about the point of execution of simple harmonic motion. This maximum displacement is termed as Amplitude.
Now, we know the differential of displacement of a particle with respect to time gives us the particle’s velocity. Thus, on differentiating equation number (1) with respect to time, we get:
$\begin{align}
& \Rightarrow \dfrac{dX}{dt}=\dfrac{d[A\sin (\omega t+\theta )]}{dt} \\
& \Rightarrow v=A\omega \cos (\omega t+\theta ) \\
\end{align}$
Here, we can write the term obtained for velocity as:
$\Rightarrow A\omega \cos (\omega t+\theta )=-A\omega \sin (\omega t+\theta +\dfrac{\pi }{2})$ [Using: $\cos \left( \dfrac{\pi }{2}+x \right)=-\sin x$ ]
$\therefore v=-A\omega \sin (\omega t+\theta +\dfrac{\pi }{2})$ [Let this expression be equation number (2)]
On comparing equation number (1) and (2), we get:
The difference in the phase of velocity and displacement as $\dfrac{\pi }{2}$ .
Therefore, for a particle executing Simple Harmonic Motion, the phase difference between velocity and displacement is equal to $\dfrac{\pi }{2}$ radian.
So, the correct answer is “Option A”.
Note: We see that the difference in phase of velocity and displacement is independent of any parameter. That is, at any given instant this difference remains the same. If we differentiate the velocity equation again, we will see that the phase difference between displacement and acceleration is $\pi $ radian.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
10 examples of evaporation in daily life with explanations
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Why is there a time difference of about 5 hours between class 10 social science CBSE