A simple harmonic oscillation is represented by the equation. `y=0.40sin(440t+0.61)`
Here y and t are in m are s respectively. What are the values of (i) amplitude (ii) angular frequency (iii) frequency of oscillations (iv) time period of oscillations and (v) initial phase?

Answer

To solve the problem, we will analyze the given equation of simple harmonic motion (SHM) and extract the required values step by step.

The equation provided is:
\[ y = 0.40 \sin(440t + 0.61) \]

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

From the equation:
\[ A = 0.40 \, \text{m} \]

### Step 2: Identify the Angular Frequency
The angular frequency \( \omega \) is the coefficient of \( t \) in the sine function.

From the equation:
\[ \omega = 440 \, \text{rad/s} \]

### Step 3: Calculate the Frequency of Oscillation
The frequency \( f \) can be calculated from the angular frequency using the formula:
\[ f = \frac{\omega}{2\pi} \]

Substituting the value of \( \omega \):
\[ f = \frac{440}{2\pi} \approx \frac{440}{6.2832} \approx 70.0 \, \text{Hz} \]

### Step 4: Calculate the Time Period of Oscillation
The time period \( T \) is the reciprocal of the frequency:
\[ T = \frac{1}{f} \]

Substituting the value of \( f \):
\[ T = \frac{1}{70} \approx 0.0143 \, \text{s} \]

### Step 5: Identify the Initial Phase
The initial phase \( \phi \) is the constant term added to \( \omega t \) in the sine function.

From the equation:
\[ \phi = 0.61 \, \text{radians} \]

### Summary of Results
1. Amplitude \( A = 0.40 \, \text{m} \)
2. Angular frequency \( \omega = 440 \, \text{rad/s} \)
3. Frequency \( f \approx 70.0 \, \text{Hz} \)
4. Time period \( T \approx 0.0143 \, \text{s} \)
5. Initial phase \( \phi = 0.61 \, \text{radians} \)