A particle executes SHM along a straight line so that its period is 12 s. The time it takes in traversing a distance equal to half its amplitude from its equilibrium position is

Options

A. 6s
B. 4s
C. 2s
D. 1s

Explanation

To solve the problem, we need to determine the time it takes for a particle executing Simple Harmonic Motion (SHM) to traverse a distance equal to half its amplitude from its equilibrium position. Here’s a step-by-step solution:

### Step 1: Understand the SHM Equation
The position of a particle in SHM can be described by the equation:
\[ x(t) = A \sin(\omega t) \]
where:
- \( x(t) \) is the displacement from the equilibrium position,
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( t \) is the time.

### Step 2: Determine the Angular Frequency
The angular frequency \( \omega \) is related to the period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
Given that the period \( T \) is 12 seconds, we can calculate \( \omega \):
\[ \omega = \frac{2\pi}{12} = \frac{\pi}{6} \, \text{rad/s} \]

### Step 3: Set Up the Equation for Half Amplitude
We want to find the time \( t \) when the particle is at half its amplitude:
\[ x = \frac{A}{2} \]
Substituting this into the SHM equation gives:
\[ \frac{A}{2} = A \sin(\omega t) \]

### Step 4: Simplify the Equation
Dividing both sides by \( A \) (assuming \( A \neq 0 \)):
\[ \frac{1}{2} = \sin(\omega t) \]

### Step 5: Solve for \( \omega t \)
The sine function equals \( \frac{1}{2} \) at:
\[ \omega t = \frac{\pi}{6} \]
Thus, we can write:
\[ \frac{\pi}{6} = \frac{\pi}{6} t \]

### Step 6: Substitute for \( \omega \)
Substituting \( \omega = \frac{\pi}{6} \):
\[ \frac{\pi}{6} = \frac{\pi}{6} t \]
Cancelling \( \pi \) from both sides:
\[ \frac{1}{6} = t \]

### Step 7: Calculate Time
Now, substituting \( \omega \) back in gives:
\[ t = \frac{1}{6} \times 12 = 1 \, \text{second} \]

### Conclusion
The time it takes for the particle to traverse a distance equal to half its amplitude from its equilibrium position is:
\[ t = 1 \, \text{second} \]

### Final Answer
The correct answer is \( t = 1 \, \text{second} \).

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