The acceleration of a certain simple harmonic oscillator is given by
`a=-(35.28 m//s^(2))cos4.2t`
The amplitude of the simple harmonic motion is

Options

A. 2.0 m
B. 8.4 m
C. 16.8 m
D. 17.64 m

Answer

To find the amplitude of the simple harmonic motion from the given acceleration equation, we can follow these steps:

### Step 1: Understand the given acceleration equation
The acceleration of the simple harmonic oscillator is given by:
\[ a = -35.28 \, \text{m/s}^2 \cos(4.2t) \]

### Step 2: Relate acceleration to displacement
In simple harmonic motion (SHM), the acceleration \( a \) can be expressed in terms of the amplitude \( A \) and angular frequency \( \omega \) as:
\[ a = -\omega^2 A \cos(\omega t) \]

### Step 3: Identify the angular frequency
From the given equation, we can identify:
\[ \omega = 4.2 \, \text{rad/s} \]

### Step 4: Compare the two equations
From the standard form of acceleration in SHM:
\[ a = -\omega^2 A \cos(\omega t) \]
we can see that:
\[ -35.28 = -\omega^2 A \]
Thus, we can write:
\[ 35.28 = \omega^2 A \]

### Step 5: Substitute the value of \( \omega \)
Now, substituting \( \omega = 4.2 \):
\[ 35.28 = (4.2)^2 A \]

### Step 6: Calculate \( \omega^2 \)
Calculating \( (4.2)^2 \):
\[ (4.2)^2 = 17.64 \]

### Step 7: Substitute and solve for \( A \)
Now, substituting this back into the equation:
\[ 35.28 = 17.64 A \]
To find \( A \), rearranging gives:
\[ A = \frac{35.28}{17.64} \]

### Step 8: Perform the division
Calculating \( A \):
\[ A = 2 \, \text{m} \]

### Conclusion
Thus, the amplitude of the simple harmonic motion is:
\[ A = 2 \, \text{m} \]