To find the distance travelled by the particle in the first three seconds, we can follow these steps:
### Step 1: Identify the parameters from the given equation
The equation of motion for the particle in Simple Harmonic Motion (SHM) is given as:
\[ x = 4 \, \text{cm} \cos\left(\frac{\pi}{2} t\right) \]
From this equation, we can identify:
- Amplitude \( A = 4 \, \text{cm} \)
- Angular frequency \( \omega = \frac{\pi}{2} \, \text{rad/s} \)
### Step 2: Determine the period of the motion
The period \( T \) of SHM can be calculated using the formula:
\[ T = \frac{2\pi}{\omega} \]
Substituting the value of \( \omega \):
\[ T = \frac{2\pi}{\frac{\pi}{2}} = 4 \, \text{s} \]
### Step 3: Analyze the motion within the first three seconds
Since the period \( T \) is 4 seconds, the particle will complete one full oscillation (from \( +A \) to \( -A \) and back to \( 0 \)) in 4 seconds. In the first 3 seconds, the particle will not complete a full oscillation.
### Step 4: Calculate the position of the particle at \( t = 3 \, \text{s} \)
We can find the position of the particle at \( t = 3 \, \text{s} \):
\[ x(3) = 4 \cos\left(\frac{\pi}{2} \times 3\right) \]
\[ x(3) = 4 \cos\left(\frac{3\pi}{2}\right) \]
Since \( \cos\left(\frac{3\pi}{2}\right) = 0 \):
\[ x(3) = 4 \times 0 = 0 \, \text{cm} \]
### Step 5: Determine the distance travelled
The particle starts from its maximum position \( +4 \, \text{cm} \) at \( t = 0 \, \text{s} \) and moves to \( 0 \, \text{cm} \) at \( t = 3 \, \text{s} \). The distance travelled by the particle is:
- From \( +4 \, \text{cm} \) to \( 0 \, \text{cm} \): \( 4 \, \text{cm} \)
In the first 3 seconds, the particle moves from \( +4 \, \text{cm} \) to \( -4 \, \text{cm} \) and then back to \( 0 \, \text{cm} \), covering:
- From \( +4 \, \text{cm} \) to \( -4 \, \text{cm} \): \( 8 \, \text{cm} \)
- From \( -4 \, \text{cm} \) to \( 0 \, \text{cm} \): \( 4 \, \text{cm} \)
Thus, total distance travelled:
\[ \text{Total distance} = 4 + 4 = 12 \, \text{cm} \]
### Final Answer
Hence, the distance travelled by the particle in the first 3 seconds is:
\[ \text{Distance} = 12 \, \text{cm} \]
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