A simple harmonic motion along the x-axis has the following properties: amplitude `=0.5m` the time to go from one extreme position to other is `2s` and `x=0.3m` at `t=0.5`. the general equation of the simple harmonic motion is

Options

A. `x=(0.5m)sin[(pit)/(2)+8^@]`
B. `x=(0.5m)sin[(pit)/(2)-8^@]`
C. `x=(0.5m)cos[(pit)/(2)+8^@]`
D. `x=(0.5m)cos[(pit)/(2)-8^@]`

Explanation

To find the general equation of the simple harmonic motion (SHM) given the properties, we will follow these steps:

### Step 1: Identify the given parameters
- Amplitude \( A = 0.5 \, \text{m} \)
- Time to go from one extreme position to the other \( = 2 \, \text{s} \)
- Position \( x = 0.3 \, \text{m} \) at \( t = 0.5 \, \text{s} \)

### Step 2: Calculate the time period \( T \)
The time to go from one extreme position to the other is half the time period. Therefore, the time period \( T \) is:
\[ T = 2 \times 2 \, \text{s} = 4 \, \text{s} \]

### Step 3: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of \( T \):
\[ \omega = \frac{2\pi}{4} = \frac{\pi}{2} \, \text{rad/s} \]

### Step 4: Write the general equation of SHM
The general equation for SHM can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
Substituting the known values:
\[ x(t) = 0.5 \sin\left(\frac{\pi}{2} t + \phi\right) \]

### Step 5: Use the given position to find \( \phi \)
We know that at \( t = 0.5 \, \text{s} \), \( x = 0.3 \, \text{m} \). Substitute these values into the equation:
\[ 0.3 = 0.5 \sin\left(\frac{\pi}{2} \times 0.5 + \phi\right) \]
This simplifies to:
\[ 0.3 = 0.5 \sin\left(\frac{\pi}{4} + \phi\right) \]
Dividing both sides by 0.5:
\[ 0.6 = \sin\left(\frac{\pi}{4} + \phi\right) \]

### Step 6: Solve for \( \phi \)
To find \( \phi \), we take the inverse sine:
\[ \frac{\pi}{4} + \phi = \sin^{-1}(0.6) \]
Calculating \( \sin^{-1}(0.6) \):
\[ \sin^{-1}(0.6) \approx 0.6435 \, \text{radians} \]
Now, solving for \( \phi \):
\[ \phi = 0.6435 - \frac{\pi}{4} \]
Calculating \( \frac{\pi}{4} \approx 0.7854 \):
\[ \phi \approx 0.6435 - 0.7854 \approx -0.1419 \, \text{radians} \approx -8.13^\circ \]

### Step 7: Write the final equation
Substituting \( \phi \) back into the general equation:
\[ x(t) = 0.5 \sin\left(\frac{\pi}{2} t - 0.1419\right) \]

### Final Answer
The general equation of the simple harmonic motion is:
\[ x(t) = 0.5 \sin\left(\frac{\pi}{2} t - 0.1419\right) \]

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