Displacement time equation of a particle executing `SHM` is, `x = 10 sin ((pi)/(3)t+(pi)/(6))cm`. The distance covered by particle in `3s` is

Options

A. `5cm`
B. `20cm`
C. `10cm`
D. `15cm`

Explanation

To find the distance covered by the particle executing simple harmonic motion (SHM) in 3 seconds, we can follow these steps:

### Step 1: Identify the displacement equation
The displacement-time equation of the particle is given as:
\[ x = 10 \sin\left(\frac{\pi}{3}t + \frac{\pi}{6}\right) \text{ cm} \]

### Step 2: Determine the angular frequency (ω)
From the equation, we can identify the angular frequency (ω):
\[ \omega = \frac{\pi}{3} \text{ rad/s} \]

### Step 3: Calculate the time period (T)
The time period (T) of the SHM can be calculated using the formula:
\[ T = \frac{2\pi}{\omega} \]
Substituting the value of ω:
\[ T = \frac{2\pi}{\frac{\pi}{3}} = 6 \text{ seconds} \]

### Step 4: Analyze the motion over 3 seconds
Since the time period is 6 seconds, 3 seconds corresponds to half of the time period:
\[ \frac{T}{2} = 3 \text{ seconds} \]

### Step 5: Determine the distance traveled in half the time period
In one complete time period (6 seconds), the particle travels from the maximum displacement on one side to the maximum displacement on the other side and back to the mean position. The total distance traveled in one complete cycle is:
- From maximum displacement to mean position: 10 cm
- From mean position to maximum displacement on the opposite side: 10 cm
- From maximum displacement back to mean position: 10 cm
- From mean position back to the original maximum displacement: 10 cm

Thus, the total distance traveled in one complete cycle is:
\[ 10 + 10 + 10 + 10 = 40 \text{ cm} \]

Since 3 seconds is half of the time period, the distance traveled in this time is half of the total distance:
\[ \text{Distance in 3 seconds} = \frac{40}{2} = 20 \text{ cm} \]

### Final Answer
The distance covered by the particle in 3 seconds is:
\[ \boxed{20 \text{ cm}} \]

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