A harmonic oscillation is represented by y=0.34 cos (3000t+0.74), where y and t are in mm and s respectively. Deduce (i) and amplitude (ii) the frequency and angular frequency (iii) the period and (iv) the intial phase.

Answer

To solve the given problem step by step, we will analyze the equation of harmonic oscillation provided:

**Given Equation:**
\[ y = 0.34 \cos(3000t + 0.74) \]

### Step 1: Identify the Amplitude
The standard form of the equation for harmonic motion is:
\[ y = A \cos(\omega t + \phi) \]

Where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the initial phase.

From the given equation, we can see that:
- Amplitude \( A = 0.34 \) mm.

**Solution for Amplitude:**
\[ \text{Amplitude} = 0.34 \, \text{mm} \]

### Step 2: Determine the Angular Frequency and Frequency
From the equation, we identify the angular frequency \( \omega \):
- \( \omega = 3000 \, \text{rad/s} \)

To find the frequency \( f \), we use the relationship:
\[ \omega = 2\pi f \]

Rearranging gives:
\[ f = \frac{\omega}{2\pi} \]

Substituting the value of \( \omega \):
\[ f = \frac{3000}{2\pi} \]

Calculating this gives:
\[ f \approx \frac{3000}{6.2832} \approx 477.46 \, \text{Hz} \]

**Solution for Frequency:**
\[ \text{Frequency} \approx 477.46 \, \text{Hz} \]

### Step 3: Calculate the Time Period
The time period \( T \) is the reciprocal of the frequency:
\[ T = \frac{1}{f} \]

Substituting the value of \( f \):
\[ T \approx \frac{1}{477.46} \]

Calculating this gives:
\[ T \approx 0.00209 \, \text{s} \]

**Solution for Time Period:**
\[ \text{Time Period} \approx 0.00209 \, \text{s} \]

### Step 4: Find the Initial Phase
From the equation, the initial phase \( \phi \) is:
- \( \phi = 0.74 \) radians.

**Solution for Initial Phase:**
\[ \text{Initial Phase} = 0.74 \, \text{radians} \]

### Summary of Results
1. **Amplitude:** \( 0.34 \, \text{mm} \)
2. **Frequency:** \( \approx 477.46 \, \text{Hz} \)
3. **Time Period:** \( \approx 0.00209 \, \text{s} \)
4. **Initial Phase:** \( 0.74 \, \text{radians} \)

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