The differnetial equation of S.H.M is given by `(d^2x)/(dt^2)+propx=0`. The frequency of motion is

Options

A. `(sqrtprop)/(2pi)`
B. `pi/(sqrtprop)`
C. `(2pi)/(sqrtprop)`
D. `prop/(2pi)`

Answer

To find the frequency of motion given the differential equation of simple harmonic motion (S.H.M), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Differential Equation**:
The given differential equation is:
\[ \frac{d^2x}{dt^2} + \alpha x = 0 \]
This is a standard form of the differential equation for simple harmonic motion.

2. **Compare with Standard Form**:
The standard form of the differential equation for S.H.M is:
\[ \frac{d^2x}{dt^2} + \omega^2 x = 0 \]
By comparing both equations, we can see that:
\[ \omega^2 = \alpha \]
where \(\omega\) is the angular frequency.

3. **Calculate Angular Frequency**:
From the comparison, we can express \(\omega\) as:
\[ \omega = \sqrt{\alpha} \]

4. **Relate Angular Frequency to Frequency**:
The frequency \(f\) is related to the angular frequency \(\omega\) by the formula:
\[ f = \frac{\omega}{2\pi} \]

5. **Substitute the Value of \(\omega\)**:
Now substituting \(\omega = \sqrt{\alpha}\) into the frequency formula:
\[ f = \frac{\sqrt{\alpha}}{2\pi} \]

6. **Final Result**:
Therefore, the frequency of motion is:
\[ f = \frac{\sqrt{\alpha}}{2\pi} \quad \text{(in Hz or s}^{-1}\text{)} \]