# 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

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**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.

Last updated date: 29th May 2023

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