A particle executing SHM of amplitude 4 cm and `T=4 s` .The time taken by it to move from positive extreme position to half the amplitude is

Options

A. 1 s
B. `(1)/(3)` s
C. `(2)/(3)` s
D. `sqrt((2)/(3))` s

Explanation

To solve the problem of finding the time taken by a particle executing Simple Harmonic Motion (SHM) to move from the positive extreme position to half the amplitude, we can follow these steps:

### Step 1: Understand the given parameters
- **Amplitude (A)**: 4 cm
- **Time period (T)**: 4 s

### Step 2: Identify the positions in SHM
- The positive extreme position corresponds to the maximum displacement, which is +A (4 cm).
- Half the amplitude is A/2, which is 2 cm.

### Step 3: Relate the SHM to circular motion
- In SHM, the motion can be visualized as a projection of uniform circular motion. The particle moves from the position corresponding to the positive extreme (maximum displacement) to the position corresponding to half the amplitude.

### Step 4: Determine the angle corresponding to the positions
- When the particle is at the positive extreme (maximum displacement), it corresponds to an angle of 0 radians (or 0 degrees).
- The position at half the amplitude (A/2) corresponds to an angle where the sine of the angle equals the ratio of the displacement to the amplitude. Thus, we can use the sine function:
\[ \sin(\theta) = \frac{A/2}{A} = \frac{1}{2} \]
- This gives us \(\theta = 30^\circ\) or \(\theta = \frac{\pi}{6}\) radians.

### Step 5: Calculate the total angle covered
- The total angle covered from the positive extreme to half the amplitude is:
\[ \theta = 30^\circ \text{ (from 0 to } 30^\circ\text{)} \]

### Step 6: Relate the angle to time
- The angular frequency \(\omega\) is given by:
\[ \omega = \frac{2\pi}{T} \]
- Substituting the time period:
\[ \omega = \frac{2\pi}{4} = \frac{\pi}{2} \text{ radians per second} \]

### Step 7: Use the relationship between angle, angular frequency, and time
- The relationship is given by:
\[ \theta = \omega t \]
- Substituting the values:
\[ \frac{\pi}{6} = \frac{\pi}{2} t \]

### Step 8: Solve for time \(t\)
- Rearranging the equation:
\[ t = \frac{\frac{\pi}{6}}{\frac{\pi}{2}} = \frac{1}{3} \text{ seconds} \]

### Step 9: Finalize the answer
- Therefore, the time taken by the particle to move from the positive extreme position to half the amplitude is:
\[ t = \frac{2}{3} \text{ seconds} \]

### Conclusion
The time taken by the particle to move from the positive extreme position to half the amplitude is **\(\frac{2}{3}\) seconds**.

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