For simple harmonic vibrations
`y_(1)=8cos omegat`
`y_(2)=4 cos (omegat+(pi)/(2))`
`y_(3)=2cos (omegat+pi)`
`y_(4)=cos(omegat+(3pi)/(2))` are superimposed on one another. The resulting amplitude and phase are respectively

Options

A. `sqrt(45) and tan^(-1)((1)/(2))`
B. `sqrt(45) and tan^(-1)((1)/(3))`
C. `sqrt(75) and tan^(-1)(2)`
D. `sqrt(75) and tan^(-1)((1)/(3))`

Answer

To solve the problem of finding the resulting amplitude and phase of the superimposed simple harmonic vibrations given by:

1. \( y_1 = 8 \cos(\omega t) \)
2. \( y_2 = 4 \cos\left(\omega t + \frac{\pi}{2}\right) \)
3. \( y_3 = 2 \cos\left(\omega t + \pi\right) \)
4. \( y_4 = \cos\left(\omega t + \frac{3\pi}{2}\right) \)

we will follow these steps:

### Step 1: Rewrite the equations in sine and cosine forms
- For \( y_2 \):
\[ y_2 = 4 \cos\left(\omega t + \frac{\pi}{2}\right) = 4 \left(-\sin(\omega t)\right) = -4 \sin(\omega t) \]

- For \( y_3 \):
\[ y_3 = 2 \cos\left(\omega t + \pi\right) = 2 \left(-\cos(\omega t)\right) = -2 \cos(\omega t) \]

- For \( y_4 \):
\[ y_4 = \cos\left(\omega t + \frac{3\pi}{2}\right) = \sin(\omega t) \]

Now we have:
- \( y_1 = 8 \cos(\omega t) \)
- \( y_2 = -4 \sin(\omega t) \)
- \( y_3 = -2 \cos(\omega t) \)
- \( y_4 = \sin(\omega t) \)

### Step 2: Combine the terms
Combine all the terms:
\[ y = y_1 + y_2 + y_3 + y_4 = (8 \cos(\omega t) - 2 \cos(\omega t)) + (-4 \sin(\omega t) + \sin(\omega t)) \]
\[ y = (8 - 2) \cos(\omega t) + (-4 + 1) \sin(\omega t) = 6 \cos(\omega t) - 3 \sin(\omega t) \]

### Step 3: Express in the form \( R \cos(\omega t + \phi) \)
To express \( y \) in the form \( R \cos(\omega t + \phi) \), we need to find \( R \) and \( \phi \).

Using the relationships:
\[ R = \sqrt{A^2 + B^2} \]
where \( A = 6 \) and \( B = -3 \):
\[ R = \sqrt{6^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} \]

### Step 4: Find the phase \( \phi \)
Using:
\[ \tan(\phi) = \frac{B}{A} = \frac{-3}{6} = -\frac{1}{2} \]
Thus,
\[ \phi = \tan^{-1}\left(-\frac{1}{2}\right) \]

### Final Result
The resulting amplitude and phase are:
- Amplitude \( R = \sqrt{45} \)
- Phase \( \phi = \tan^{-1}\left(-\frac{1}{2}\right) \)