A body performs a S.H.M. of amplitude a. If the speed of the particle is half of its maximum speed, then the displacement of the particle is

Options

A. `(a)/(2)`
B. `(sqrt(3))/(2)a`
C. a
D. 2a

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We are given that a body performs Simple Harmonic Motion (S.H.M.) with an amplitude \( a \). We need to find the displacement \( x \) of the particle when its speed \( v \) is half of its maximum speed \( v_{\text{max}} \).

### Step 2: Write down the formula for maximum speed
The maximum speed \( v_{\text{max}} \) in S.H.M. is given by the formula:
\[ v_{\text{max}} = \omega a \]
where \( \omega \) is the angular frequency and \( a \) is the amplitude.

### Step 3: Express the speed when it is half of the maximum speed
If the speed of the particle is half of its maximum speed, we can write:
\[ v = \frac{1}{2} v_{\text{max}} = \frac{1}{2} \omega a \]

### Step 4: Write the formula for speed in S.H.M.
The speed \( v \) of a particle in S.H.M. at displacement \( x \) is given by:
\[ v = \omega \sqrt{a^2 - x^2} \]

### Step 5: Set the two expressions for speed equal
Now we can set the two expressions for speed equal to each other:
\[ \omega \sqrt{a^2 - x^2} = \frac{1}{2} \omega a \]

### Step 6: Cancel \( \omega \) from both sides
Assuming \( \omega \neq 0 \), we can divide both sides by \( \omega \):
\[ \sqrt{a^2 - x^2} = \frac{1}{2} a \]

### Step 7: Square both sides to eliminate the square root
Squaring both sides gives:
\[ a^2 - x^2 = \left(\frac{1}{2} a\right)^2 \]
\[ a^2 - x^2 = \frac{1}{4} a^2 \]

### Step 8: Rearrange the equation to solve for \( x^2 \)
Rearranging the equation, we get:
\[ x^2 = a^2 - \frac{1}{4} a^2 \]
\[ x^2 = \frac{3}{4} a^2 \]

### Step 9: Take the square root to find \( x \)
Taking the square root of both sides gives:
\[ x = \sqrt{\frac{3}{4}} a = \frac{\sqrt{3}}{2} a \]

### Final Answer
Thus, the displacement of the particle when its speed is half of its maximum speed is:
\[ x = \frac{\sqrt{3}}{2} a \]

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