Question

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 1

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.