In the differl equation `(d^(2)x)/(dt^(2))+omega^(2)x=0` , for a simple harmonic motion, the term`omega^(2)` represents

Options

A. restoring force per unit mass
B. restoring force per unit displacement
C. restoring force per unit mass per unit displacement
D. acceleration per unit mass per unit displacemnet

Answer

To solve the question regarding the term \( \omega^2 \) in the differential equation

\[ \frac{d^2x}{dt^2} + \omega^2 x = 0 \]

for simple harmonic motion (SHM), we can follow these steps:

### Step 1: Understand the context of SHM
In simple harmonic motion, an object oscillates back and forth around an equilibrium position. The restoring force acting on the object is proportional to the displacement from the equilibrium position and acts in the opposite direction.

### Step 2: Recall the relationship between force, mass, and acceleration
According to Newton's second law, the force \( F \) acting on an object is given by:

\[ F = m \cdot a \]

where \( m \) is the mass of the object and \( a \) is its acceleration. In SHM, the restoring force \( F \) can be expressed as:

\[ F = -k \cdot x \]

where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

### Step 3: Relate force to acceleration
The acceleration \( a \) can be expressed as the second derivative of displacement with respect to time:

\[ a = \frac{d^2x}{dt^2} \]

Substituting this into Newton's second law gives us:

\[ -k \cdot x = m \cdot \frac{d^2x}{dt^2} \]

### Step 4: Rearranging the equation
Rearranging the above equation leads to:

\[ \frac{d^2x}{dt^2} + \frac{k}{m} x = 0 \]

### Step 5: Identify \( \omega^2 \)
By comparing this equation with the original differential equation \( \frac{d^2x}{dt^2} + \omega^2 x = 0 \), we can identify that:

\[ \omega^2 = \frac{k}{m} \]

### Step 6: Interpret \( \omega^2 \)
The term \( \omega^2 \) represents the angular frequency squared of the oscillation. It is a measure of how quickly the oscillation occurs and is related to the physical properties of the system (specifically, the spring constant \( k \) and the mass \( m \)).

### Conclusion
Thus, in the context of the question, the term \( \omega^2 \) represents:

\[ \omega^2 = \frac{k}{m} \]

This indicates that \( \omega^2 \) is the ratio of the spring constant to the mass, which is crucial in determining the characteristics of the simple harmonic motion.

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