The equation of a particle executing `SHM` is `x = (5m)sin[(pis^(-1))t + (pi)/(6)]`. Write down the amplitude, initial phase constant, time period and maximum speed.

Answer

To solve the problem, we need to analyze the given equation of motion for a particle executing Simple Harmonic Motion (SHM):

**Given equation:**
\[ x = 5 \sin\left(\pi t + \frac{\pi}{6}\right) \]

### Step 1: Identify the Amplitude
The amplitude \( A \) is the coefficient of the sine function in the SHM equation.

**Solution:**
From the equation, we can see that:
\[ A = 5 \, \text{m} \]

### Step 2: Identify the Initial Phase Constant
The initial phase constant \( \phi \) is the constant added to the argument of the sine function.

**Solution:**
From the equation, we have:
\[ \phi = \frac{\pi}{6} \]

### Step 3: Calculate the Time Period
The time period \( T \) of SHM is related to the angular frequency \( \omega \) by the formula:
\[ T = \frac{2\pi}{\omega} \]

**Solution:**
From the equation, we can identify \( \omega \) (the coefficient of \( t \)):
\[ \omega = \pi \, \text{rad/s} \]
Now substituting this value into the time period formula:
\[ T = \frac{2\pi}{\pi} = 2 \, \text{s} \]

### Step 4: Calculate the Maximum Speed
The maximum speed \( V_{\text{max}} \) of a particle in SHM is given by the formula:
\[ V_{\text{max}} = A \omega \]

**Solution:**
Substituting the values of \( A \) and \( \omega \):
\[ V_{\text{max}} = 5 \times \pi = 5\pi \, \text{m/s} \]

### Final Summary of Results:
- Amplitude \( A = 5 \, \text{m} \)
- Initial Phase Constant \( \phi = \frac{\pi}{6} \)
- Time Period \( T = 2 \, \text{s} \)
- Maximum Speed \( V_{\text{max}} = 5\pi \, \text{m/s} \)

---