Question

Which of the following equations represents a simple harmonic motion? (\(x\) is displacement and \(a\) is acceleration)

• A. \(a = 0.9x^2\)
• B. \(a = -10x\)
• C. \(a = 100x^3\)
• D. \(a = 1.2x\)
• E. \(a^2 = 12x\)

Hint:

In SHM, the "negative sign" is just as important as the "linear x". Without the negative sign, the motion is unstable and exponential, not oscillatory.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Simple Harmonic Motion (SHM) is a periodic motion where the restoring force (and thus acceleration) is directly proportional to the displacement and directed towards the equilibrium position.

Step 2: Key Formula or Approach:
The defining equation of SHM is \(a = -\omega^2 x\), where \(\omega\) is the angular frequency.

Step 3: Detailed Explanation:
1. Condition 1: Acceleration must be linearly proportional to displacement (\(a \propto x\)). This eliminates options A, C, and E.
2. Condition 2: Acceleration must act in the opposite direction to displacement (the negative sign). This eliminates option D.
3. Checking Option B: \(a = -10x\). This fits the form \(a = -\omega^2 x\) with \(\omega = \sqrt{10}\).

Step 4: Final Answer:
The equation \(a = -10x\) represents SHM.