A particle is performing SHM along x-axis with amplitude 6.0 cm and time period 12 s. What is the minimum time taken by the particle to move from `x=+3cm` to `x=+6cm` and back again?

Options

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

Answer

To solve the problem of a particle performing Simple Harmonic Motion (SHM) along the x-axis with an amplitude of 6.0 cm and a time period of 12 seconds, we need to find the minimum time taken by the particle to move from \(x = +3 \, \text{cm}\) to \(x = +6 \, \text{cm}\) and back again.

### Step-by-Step Solution:

1. **Identify the parameters of SHM:**
- Amplitude \(A = 6 \, \text{cm}\)
- Time period \(T = 12 \, \text{s}\)

2. **Calculate the angular frequency \(\omega\):**
\[ \omega = \frac{2\pi}{T} = \frac{2\pi}{12} = \frac{\pi}{6} \, \text{rad/s} \]

3. **Use the SHM displacement equation:**
The displacement \(x\) in SHM is given by:
\[ x = A \sin(\omega t) \]
We need to find the times \(t_1\) and \(t_2\) when the particle is at \(x = 3 \, \text{cm}\) and \(x = 6 \, \text{cm}\), respectively.

4. **Find \(t_1\) when \(x = 3 \, \text{cm}\):**
\[ 3 = 6 \sin\left(\frac{\pi}{6} t_1\right) \]
Simplifying gives:
\[ \sin\left(\frac{\pi}{6} t_1\right) = \frac{3}{6} = \frac{1}{2} \]
The angle whose sine is \(\frac{1}{2}\) is \(\frac{\pi}{6}\) (1st quadrant). Thus:
\[ \frac{\pi}{6} t_1 = \frac{\pi}{6} \implies t_1 = 1 \, \text{s} \]

5. **Find \(t_2\) when \(x = 6 \, \text{cm}\):**
\[ 6 = 6 \sin\left(\frac{\pi}{6} t_2\right) \]
Simplifying gives:
\[ \sin\left(\frac{\pi}{6} t_2\right) = 1 \]
The angle whose sine is \(1\) is \(\frac{\pi}{2}\). Thus:
\[ \frac{\pi}{6} t_2 = \frac{\pi}{2} \implies t_2 = 3 \, \text{s} \]

6. **Calculate the time taken to move from \(x = 3 \, \text{cm}\) to \(x = 6 \, \text{cm}\):**
\[ \Delta t = t_2 - t_1 = 3 \, \text{s} - 1 \, \text{s} = 2 \, \text{s} \]

7. **Calculate the total time for the complete motion (to \(x = 6 \, \text{cm}\) and back to \(x = 3 \, \text{cm}\)):**
\[ \text{Total time} = 2 \times \Delta t = 2 \times 2 \, \text{s} = 4 \, \text{s} \]

### Final Answer:
The minimum time taken by the particle to move from \(x = +3 \, \text{cm}\) to \(x = +6 \, \text{cm}\) and back again is **4 seconds**.