Question

In the differential equation of linear simple harmonic motion \(\displaystyle \frac{d^2x}{dt^2} + \omega^2 x = 0\), the term \( \omega^2 \) represents

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

Hint:

In SHM, always remember: \[ \omega^2 = \frac{k}{m} \] which directly shows restoring force per unit mass per unit displacement.

Answer 1

The Correct Option is A

Step 1: Writing the SHM equation.
The standard equation of linear simple harmonic motion is:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \]
Step 2: Interpreting the terms.
Acceleration is proportional to displacement and acts in the opposite direction. Hence:
\[ \text{Acceleration} = -(\text{constant}) \times \text{displacement} \]
Step 3: Physical meaning of \( \omega^2 \).
Since acceleration is force per unit mass, the constant \( \omega^2 \) represents the restoring force per unit mass per unit displacement.
Step 4: Conclusion.
Therefore, \( \omega^2 \) denotes restoring force per unit mass per unit displacement.