Answer
64.8k+ views
Hint:In this question, describe the simple harmonic motion and then find out whether $\sin wt - \cos wt$ can be rewritten as a mathematical expression of a simple harmonic motion and then find out the period the mass takes to complete its oscillation.
Complete step by step solution:
In the question, we have given a function that is, $\sin wt - \cos wt$
Now, we can rewrite the given function as
$\sin wt - \cos wt = \sqrt 2 \left[ {\dfrac{1}{{\sqrt 2 }}\sin wt - \dfrac{1}{{\sqrt 2 }}\cos wt} \right]$
We can write the above function as,
$ \Rightarrow \sin wt - \cos wt = \sqrt 2 \left[ {\sin wt \cdot \cos \dfrac{\pi }{4} - \cos wt \cdot \sin \dfrac{\pi }{4}} \right]$
After simplification, we can write it as,
$ \Rightarrow \sin wt - \cos wt = \sqrt 2 \sin \left( {wt - \dfrac{\pi }{4}} \right)$
A simple harmonic motion is a periodic motion where the restoring force is directly proportional to the magnitude of displacement and it acts towards the equilibrium state.
The mathematical representation of a simple harmonic motion can be written as, $y = A\sin wt \pm \phi $
Where$A$is the maximum displacement of a particle from its equilibrium,$w$is the angular frequency in radians per second.
So, $\sqrt 2 \sin \left( {wt - \dfrac{\pi }{4}} \right)$ is in the form of $y = A\sin wt \pm \phi $, hence we can say it’s a simple harmonic motion.
Now the period of the motion is $\dfrac{{2\pi }}{w}$ as the time it takes to move from $A$to$ - A$and come back again is the time it takes for$wt$to advance by $2\pi $.
Hence, $wT = 2\pi \Rightarrow T = \dfrac{{2\pi }}{w}$
Therefore, the period it takes to move is $\dfrac{{2\pi }}{w}$.
Thus, we can say $\sin wt - \cos wt$represents a simple harmonic motion with a period $\dfrac{{2\pi }}{w}$.
Hence option (B) is the correct answer.
Note:The motion is actually called harmonic because musical instruments make corresponding sound waves in air. The combination of many simple harmonic motions mainly produces musical sounds.
Complete step by step solution:
In the question, we have given a function that is, $\sin wt - \cos wt$
Now, we can rewrite the given function as
$\sin wt - \cos wt = \sqrt 2 \left[ {\dfrac{1}{{\sqrt 2 }}\sin wt - \dfrac{1}{{\sqrt 2 }}\cos wt} \right]$
We can write the above function as,
$ \Rightarrow \sin wt - \cos wt = \sqrt 2 \left[ {\sin wt \cdot \cos \dfrac{\pi }{4} - \cos wt \cdot \sin \dfrac{\pi }{4}} \right]$
After simplification, we can write it as,
$ \Rightarrow \sin wt - \cos wt = \sqrt 2 \sin \left( {wt - \dfrac{\pi }{4}} \right)$
A simple harmonic motion is a periodic motion where the restoring force is directly proportional to the magnitude of displacement and it acts towards the equilibrium state.
The mathematical representation of a simple harmonic motion can be written as, $y = A\sin wt \pm \phi $
Where$A$is the maximum displacement of a particle from its equilibrium,$w$is the angular frequency in radians per second.
So, $\sqrt 2 \sin \left( {wt - \dfrac{\pi }{4}} \right)$ is in the form of $y = A\sin wt \pm \phi $, hence we can say it’s a simple harmonic motion.
Now the period of the motion is $\dfrac{{2\pi }}{w}$ as the time it takes to move from $A$to$ - A$and come back again is the time it takes for$wt$to advance by $2\pi $.
Hence, $wT = 2\pi \Rightarrow T = \dfrac{{2\pi }}{w}$
Therefore, the period it takes to move is $\dfrac{{2\pi }}{w}$.
Thus, we can say $\sin wt - \cos wt$represents a simple harmonic motion with a period $\dfrac{{2\pi }}{w}$.
Hence option (B) is the correct answer.
Note:The motion is actually called harmonic because musical instruments make corresponding sound waves in air. The combination of many simple harmonic motions mainly produces musical sounds.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)