Which of the following relationships between the acceleration a and the displacement x of a particle involve simple harmonic motion?
Explanation
For a motion to be simple harmonic, the acceleration must be proportional to displacement and directed towards the centre (restoring force). From the context, the defining condition for SHM is a(t) = -ω²x(t), where acceleration is linearly proportional to displacement with a negative sign. Among the given options: (a) has positive sign (acceleration away from centre), (b) has x² (non-linear), (c) has a = -10x which matches the form a = -ω²x with ω² = 10, and (d) has x³ (non-linear) with positive sign. Only option (c) satisfies the necessary condition for SHM.
Solution Steps
Step 1: Recall the condition for SHM from context: a(t) = -ω²x(t), meaning acceleration must be linearly proportional to displacement with negative sign.
Step 2: Check each option for linearity and sign: (a) a = 0.7x has wrong sign, (b) a = -200x² is non-linear, (c) a = -10x matches SHM form, (d) a = 100x³ is non-linear with wrong sign.
Step 3: Conclude that only option (c) a = -10x represents simple harmonic motion.