A partilce is executive simple harmonic motion given by
`x=5sin(4t-pi/6)`
The velocity of the particle when its displacement is 3 units is

Options

A. `(2pi)/(3)` units
B. `(5pi)/(6)` units
C. `20` units
D. `16` units

Explanation

To find the velocity of a particle executing simple harmonic motion given by the equation \( x = 5 \sin(4t - \frac{\pi}{6}) \) when its displacement is 3 units, we can follow these steps:

### Step 1: Identify the parameters from the equation
The equation of motion is given in the form \( x = a \sin(\omega t + \phi) \). Here:
- \( a = 5 \) (amplitude)
- \( \omega = 4 \) (angular frequency)
- \( \phi = -\frac{\pi}{6} \) (phase constant)

### Step 2: Use the velocity formula
The velocity \( v \) of a particle in simple harmonic motion can be calculated using the formula:
\[ v = \omega \sqrt{a^2 - x^2} \]
where \( x \) is the displacement at which we want to find the velocity.

### Step 3: Substitute the known values
We know:
- \( \omega = 4 \)
- \( a = 5 \)
- \( x = 3 \)

Now, substituting these values into the velocity formula:
\[ v = 4 \sqrt{5^2 - 3^2} \]

### Step 4: Calculate \( a^2 - x^2 \)
Calculate \( a^2 \) and \( x^2 \):
- \( a^2 = 5^2 = 25 \)
- \( x^2 = 3^2 = 9 \)

Now, substitute these into the equation:
\[ v = 4 \sqrt{25 - 9} \]

### Step 5: Simplify the expression
Calculate \( 25 - 9 \):
\[ 25 - 9 = 16 \]
So, we have:
\[ v = 4 \sqrt{16} \]

### Step 6: Find the square root
Calculate \( \sqrt{16} \):
\[ \sqrt{16} = 4 \]
Thus, substituting back, we get:
\[ v = 4 \times 4 = 16 \]

### Final Answer
The velocity of the particle when its displacement is 3 units is \( 16 \) units.

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