Question 140: Waves

Question

For a periodic motion represented by the equation
$$y=\sin \omega \mathrm{t}+\cos \omega \mathrm{t}$$
the amplitude of the motion is

Accepted Answer

We can write the given equation as:
$$y = \sqrt{(\sin\omega t)^2 + (\cos\omega t)^2} \cos\left(\omega t - \arctan\frac{\sin\omega t}{\cos\omega t}ight)$$
Using the identity $\sin^2	heta + \cos^2	heta = 1$, we get:
$$y = \sqrt{1 + \sin 2\omega t} \cos\left(\omega t - \frac{\pi}{4}ight)$$
The amplitude of the motion is the maximum value of $|y|$, which occurs when $\sin 2\omega t = 1$, i.e., at $t = \frac{\pi}{4\omega} + \frac{n\pi}{\omega}$, where $n$ is an integer. Substituting this value of $t$ in the above equation, we get:
$$|y_{	ext{max}}| = \sqrt{1 + \sin \frac{\pi}{2}} = \sqrt{2}$$
Therefore, the amplitude of the motion is $\sqrt{2}$

Answer

1

Answer

$$\sqrt2$$

Answer

0.5

Answer

2

Waves

For a periodic motion represented by the equation $y=\sin \omega \mathrm{t}+\cos \omega \mathrm{t}$ the amplitude of the motion is

Choose the correct answer: