
The function $\sin wt - \cos wt$ represents
A) A simple harmonic motion with a period of $\dfrac{\pi }{w}$
B) A simple harmonic motion with a period $\dfrac{{2\pi }}{w}$
C) A periodic, but not simple harmonic motion with a period $\dfrac{\pi }{w}$
D) A periodic, but not simple harmonic motion with a period $\dfrac{{2\pi }}{w}$
Answer
148.2k+ 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.
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