The equation of S.H.M. is`y=a sin (2 pint + a)`, then its phase at time t is

Options

A. `2 pint`
B. a
C. `2 pint+a`
D. `3 pint`

Answer

To find the phase of the simple harmonic motion (S.H.M.) given by the equation \( y = a \sin(2 \pi n t + \phi) \), we can follow these steps:

### Step 1: Identify the given equation
The equation of S.H.M. is given as:
\[ y = a \sin(2 \pi n t + a) \]

### Step 2: Compare with the standard form
The standard form of the equation for S.H.M. is:
\[ y = a \sin(\omega t + \phi) \]
where:
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

### Step 3: Identify the components
From the given equation \( y = a \sin(2 \pi n t + a) \):
- The term \( 2 \pi n \) corresponds to \( \omega \).
- The term \( a \) corresponds to \( \phi \).

### Step 4: Write the expression for the phase
The total phase \( \Phi \) at any time \( t \) can be expressed as:
\[ \Phi = \omega t + \phi \]

### Step 5: Substitute the values
Substituting the identified values:
- \( \omega = 2 \pi n \)
- \( \phi = a \)

Thus, the phase at time \( t \) is:
\[ \Phi = (2 \pi n) t + a \]

### Final Answer
The phase at time \( t \) is:
\[ \Phi = 2 \pi n t + a \]

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