The equation of SHM of a particle is `(d^2y)/(dt^2)+ky=0`, where k is a positive constant. The time period of motion is

Options

A. `(2pi)/(sqrt(k))`
B. `(2pi)/k`
C. `2pik`
D. `2pisqrt(k)`

Explanation

To find the time period of motion for the given equation of simple harmonic motion (SHM), we can follow these steps:

### Step 1: Identify the given equation
The equation provided is:
\[ \frac{d^2y}{dt^2} + ky = 0 \]
where \( k \) is a positive constant.

### Step 2: Compare with the standard SHM equation
The standard form of the differential equation for simple harmonic motion is:
\[ \frac{d^2y}{dt^2} + \omega^2 y = 0 \]
where \( \omega \) is the angular frequency.

### Step 3: Relate the constants
By comparing the two equations, we can see that:
\[ \omega^2 = k \]
From this, we can find \( \omega \):
\[ \omega = \sqrt{k} \]

### Step 4: Use the formula for the time period
The time period \( T \) of simple harmonic motion is given by the formula:
\[ T = \frac{2\pi}{\omega} \]
Substituting the value of \( \omega \):
\[ T = \frac{2\pi}{\sqrt{k}} \]

### Final Answer
Thus, the time period of the motion is:
\[ T = \frac{2\pi}{\sqrt{k}} \text{ seconds} \]

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