At a particular position the velocity of a particle in SHM with amplitude a is `sqrt3 // 2` that at its mean position. In this position, its displacement is :

Options

A. `a//2`
B. `sqrt3 a//2`
C. `a sqrt2`
D. `sqrt 2a`

Answer

To solve the problem, we need to find the displacement of a particle in Simple Harmonic Motion (SHM) when its velocity is given as \(\frac{\sqrt{3}}{2}\) times the velocity at its mean position. Let's go through the solution step by step.

### Step 1: Understand the equations of SHM
The general equation for displacement \(x\) in SHM is given by:
\[ x = A \sin(\omega t) \]
where \(A\) is the amplitude and \(\omega\) is the angular frequency.

### Step 2: Find the expression for velocity
The velocity \(v\) in SHM can be derived by differentiating the displacement with respect to time:
\[ v = \frac{dx}{dt} = A \omega \cos(\omega t) \]

### Step 3: Determine the velocity at the mean position
At the mean position, the displacement \(x = 0\), which occurs when \(\omega t = 0\). At this point, the velocity is maximized:
\[ v_{\text{mean}} = A \omega \]

### Step 4: Relate the given velocity to the mean position velocity
According to the problem, the velocity at a certain position is:
\[ v = \frac{\sqrt{3}}{2} v_{\text{mean}} = \frac{\sqrt{3}}{2} (A \omega) \]

### Step 5: Set up the equation
From the expression for velocity, we can set up the equation:
\[ A \omega \cos(\omega t) = \frac{\sqrt{3}}{2} (A \omega) \]
Dividing both sides by \(A \omega\) (assuming \(A \neq 0\) and \(\omega \neq 0\)):
\[ \cos(\omega t) = \frac{\sqrt{3}}{2} \]

### Step 6: Solve for \(\omega t\)
The cosine function equals \(\frac{\sqrt{3}}{2}\) at:
\[ \omega t = \frac{\pi}{6} \quad \text{and} \quad \omega t = \frac{11\pi}{6} \quad (\text{in the range } [0, 2\pi]) \]

### Step 7: Find the displacement at \(\omega t = \frac{\pi}{6}\)
Now we can find the displacement \(x\) at \(\omega t = \frac{\pi}{6}\):
\[ x = A \sin\left(\frac{\pi}{6}\right) \]
Since \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\):
\[ x = A \cdot \frac{1}{2} = \frac{A}{2} \]

### Conclusion
Thus, the displacement of the particle at the given position is:
\[ \boxed{\frac{A}{2}} \]