The differential equation of a particle executing simple harmonic mation along y-aixs is

Options

A. `(d^2y)/(dt^2)+omega^2y=0`
B. `(d^2y)/(dt^2)+omega^2y^2=0`
C. `(d^2y)/(dt^2)-omega^2y=0`
D. `(dy)/(dt)+omegay=0`

Answer

To derive the differential equation of a particle executing simple harmonic motion (SHM) along the y-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding SHM**:
In simple harmonic motion, the acceleration of the particle is directly proportional to its displacement from the equilibrium position and is directed towards that position. The general formula for acceleration in SHM is given by:
\[ a = -\omega^2 x \]
where \( \omega \) is the angular frequency and \( x \) is the displacement.

2. **Applying to the y-axis**:
Since we are interested in motion along the y-axis, we replace \( x \) with \( y \). Thus, the equation becomes:
\[ a = -\omega^2 y \]

3. **Expressing acceleration**:
Acceleration \( a \) can also be expressed in terms of the second derivative of displacement with respect to time:
\[ a = \frac{d^2y}{dt^2} \]

4. **Formulating the differential equation**:
Substituting the expression for acceleration into the SHM equation gives us:
\[ \frac{d^2y}{dt^2} = -\omega^2 y \]

5. **Final form of the differential equation**:
This is the differential equation that describes the motion of a particle executing simple harmonic motion along the y-axis:
\[ \frac{d^2y}{dt^2} + \omega^2 y = 0 \]

### Final Answer:
The differential equation of a particle executing simple harmonic motion along the y-axis is:
\[ \frac{d^2y}{dt^2} + \omega^2 y = 0 \]