The restoring force F acting on a particle of mass (m) executing a S. H. M. is given by F= -Kx, where x is the displacement and K is a constant. Then the angular velocity of the particle is given by

Options

A. `sqrt((m)/(K))`
B. `sqrt((K)/(m))`
C. mK
D. `sqrt(mK)`

Explanation

To find the angular velocity of a particle executing Simple Harmonic Motion (SHM) given the restoring force \( F = -Kx \), we can follow these steps:

### Step 1: Understand the relationship between force and acceleration
The restoring force in SHM is given by:
\[ F = -Kx \]
According to Newton's second law, the force can also be expressed as:
\[ F = ma \]
where \( a \) is the acceleration of the particle.

### Step 2: Relate acceleration to displacement
In SHM, the acceleration \( a \) can be expressed in terms of displacement \( x \) as:
\[ a = \frac{d^2x}{dt^2} = -\frac{K}{m}x \]
This shows that the acceleration is directly proportional to the displacement and is directed towards the mean position.

### Step 3: Identify the angular frequency
The standard form of the equation for SHM is:
\[ \frac{d^2x}{dt^2} + \omega^2 x = 0 \]
Comparing this with our earlier expression \( a = -\frac{K}{m}x \), we can identify that:
\[ \omega^2 = \frac{K}{m} \]
Thus, the angular frequency \( \omega \) is given by:
\[ \omega = \sqrt{\frac{K}{m}} \]

### Step 4: Conclusion
The angular velocity \( \omega \) of the particle executing SHM can be expressed as:
\[ \omega = \sqrt{\frac{K}{m}} \]

### Summary
The angular velocity of the particle is given by:
\[ \omega = \sqrt{\frac{K}{m}} \]

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