If a simple harmonic motion is given by
`y=(sin omegat+cosomegat)m`
Which of the following statements are true:

Options

A. Time is considered from `y=0m`
B. Time is considered from `y=1m`
C. The amplitude is `sqrt(2)m`
D. The amplitude is `1m`

Explanation

To analyze the given simple harmonic motion (SHM) represented by the equation:

\[ y = (\sin(\omega t) + \cos(\omega t)) \, \text{m} \]

we will follow these steps:

### Step 1: Evaluate \( y \) at \( t = 0 \)

To find the displacement at \( t = 0 \):

\[ y(0) = \sin(0) + \cos(0) = 0 + 1 = 1 \, \text{m} \]

### Step 2: Find the Amplitude

The given equation can be rewritten to find the amplitude. We can express \( y \) in the form of \( R \sin(\omega t + \phi) \).

1. First, we rewrite \( \sin(\omega t) + \cos(\omega t) \):
\[ y = \sin(\omega t) + \cos(\omega t) \]

2. The amplitude \( R \) can be calculated using the formula:
\[ R = \sqrt{A^2 + B^2} \]
where \( A = 1 \) (coefficient of \( \sin(\omega t) \)) and \( B = 1 \) (coefficient of \( \cos(\omega t) \)):
\[ R = \sqrt{1^2 + 1^2} = \sqrt{2} \]

### Step 3: Rewrite the equation in the standard form

We can express the equation as:
\[ y = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin(\omega t) + \frac{1}{\sqrt{2}} \cos(\omega t) \right) \]
This can be recognized as:
\[ y = \sqrt{2} \sin\left(\omega t + \frac{\pi}{4}\right) \]

### Step 4: Identify the maximum displacement

The maximum displacement (amplitude) of the SHM is given by the coefficient of the sine function:
\[ \text{Amplitude} = \sqrt{2} \, \text{m} \]

### Conclusion

From the analysis, we can conclude:
1. At \( t = 0 \), \( y = 1 \, \text{m} \).
2. The amplitude of the motion is \( \sqrt{2} \, \text{m} \).

### Final Statements

Based on the findings:
- The statement that \( y = 0 \, \text{m} \) at \( t = 0 \) is **false**.
- The statement that \( y = 1 \, \text{m} \) at \( t = 0 \) is **true**.
- The amplitude is \( \sqrt{2} \, \text{m} \), which is **true**.