The equation of a simple harmonic motion is `X=0.34 cos (3000t+0.74)` where X and t are in mm and sec . The frequency of motion is

Options

A. 3000
B. `3000// 2 pi`
C. `0.74 // 2 pi`
D. `3000//pi`

Answer

To find the frequency of the simple harmonic motion given by the equation \( X = 0.34 \cos(3000t + 0.74) \), we can follow these steps:

### Step 1: Identify the angular frequency (\( \omega \))
The equation of simple harmonic motion can be compared to the standard form:
\[ X = A \cos(\omega t + \phi) \]
where \( \omega \) is the angular frequency. From the given equation \( X = 0.34 \cos(3000t + 0.74) \), we can see that:
\[ \omega = 3000 \, \text{rad/s} \]

### Step 2: Relate angular frequency to frequency
The relationship between angular frequency (\( \omega \)) and frequency (\( f \)) is given by the formula:
\[ \omega = 2\pi f \]
We can rearrange this formula to solve for frequency:
\[ f = \frac{\omega}{2\pi} \]

### Step 3: Substitute the value of \( \omega \)
Now, substituting the value of \( \omega \):
\[ f = \frac{3000}{2\pi} \]

### Step 4: Calculate the frequency
Now we can calculate the frequency:
\[ f \approx \frac{3000}{6.2832} \approx 477.46 \, \text{Hz} \]

### Conclusion
Thus, the frequency of the motion is approximately \( 477.46 \, \text{Hz} \).

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