The acceleration of a particle performing simple harmonic motion is 12 cm/s^2 at a distance of 3 cm from the equilibrium position. Its time period will be :

Question

Accepted Answer

To determine the time period of a particle performing simple harmonic motion (SHM), we need to use the relationship between acceleration, displacement, and angular frequency in SHM. The acceleration $ a $ of a particle in SHM is given by the formula:

$$ a = -\omega^2 x $$

where:
- $ a $ is the acceleration,
- $ \omega $ is the angular frequency,
- $ x $ is the displacement from the equilibrium position.

Given:
- $ a = 12 \, \text{cm/s}^2 $
- $ x = 3 \, \text{cm} $

Substituting these values into the formula, we have:

$$ 12 = \omega^2 \times 3 $$

Solving for $ \omega^2 $:

$$ \omega^2 = \frac{12}{3} = 4 $$

Thus, $ \omega = \sqrt{4} = 2 \, \text{rad/s} $.

The time period $ T $ of SHM is related to the angular frequency by the formula:

$$ T = \frac{2\pi}{\omega} $$

Substituting the value of $ \omega $:

$$ T = \frac{2\pi}{2} = \pi \, \text{seconds} $$

Approximating $ \pi $ as 3.14, we find:

$$ T \approx 3.14 \, \text{seconds} $$

Therefore, the correct time period of the particle is approximately 3.14 seconds, which corresponds to Option 2.

**Explanation for Other Options:**

1. **Option 1 (6.28 seconds):** This option suggests a time period that is double the correct value. Since the calculated time period using the given acceleration and displacement is approximately 3.14 seconds, 6.28 seconds is incorrect.

2. **Option 3 (2 seconds):** This option underestimates the time period. The calculation clearly shows that the time period is approximately 3.14 seconds, not 2 seconds.

3. **Option 4 (5 seconds):** This option also overestimates the time period. The correct calculation yields a time period of approximately 3.14 seconds, making 5 seconds incorrect.

In conclusion, the correct time period of the particle performing simple harmonic motion, based on the given acceleration and displacement, is approximately 3.14 seconds, which is why Option 2 is the correct answer.