If a SHM is represented by the equation `x=10 sin(pit+(pi)/(6))` in Si units, then determine its amplitude, time period and maximum velocity `upsilon_(max)` ?

Explanation

To solve the problem step by step, we will analyze the given equation of simple harmonic motion (SHM) and extract the required parameters: amplitude, time period, and maximum velocity.

### Step 1: Identify the Standard Form of SHM
The standard equation of SHM is given by:
\[ x = A \sin(\omega t + \phi) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

### Step 2: Compare with the Given Equation
The given equation is:
\[ x = 10 \sin\left(\pi t + \frac{\pi}{6}\right) \]
By comparing this with the standard form, we can identify the values of \( A \), \( \omega \), and \( \phi \):
- Amplitude \( A = 10 \) meters
- Angular frequency \( \omega = \pi \) radians/second
- Phase constant \( \phi = \frac{\pi}{6} \) radians

### Step 3: Determine the Amplitude
From the comparison, we find:
\[ \text{Amplitude} = A = 10 \, \text{meters} \]

### Step 4: Calculate the Time Period
The time period \( T \) of SHM is related to the angular frequency \( \omega \) by the formula:
\[ T = \frac{2\pi}{\omega} \]
Substituting the value of \( \omega \):
\[ T = \frac{2\pi}{\pi} = 2 \, \text{seconds} \]

### Step 5: Calculate the Maximum Velocity
The maximum velocity \( v_{\text{max}} \) in SHM is given by:
\[ v_{\text{max}} = \omega A \]
Substituting the values of \( \omega \) and \( A \):
\[ v_{\text{max}} = \pi \times 10 = 10\pi \, \text{meters/second} \]

### Final Results
1. Amplitude \( A = 10 \, \text{meters} \)
2. Time Period \( T = 2 \, \text{seconds} \)
3. Maximum Velocity \( v_{\text{max}} = 10\pi \, \text{meters/second} \)

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