Simple Harmonic Motion — Direction of Acceleration A body is said to execute simple harmonic motion (SHM) if its acceleration is directed toward the mean (equilibrium) position and proportional to displacement from it. Is the statement (direction toward mean position) correct?
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ATrue
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BFalse
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C—
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D—
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E—
Answer
Correct Answer: True
Explanation
Introduction / Context: SHM is a foundational motion model for oscillators (springs, pendulums for small angles, masses). It underpins vibration theory, acoustics, and many control-system approximations near equilibrium.
Given Data / Assumptions:
- Displacement x measured from equilibrium.
- Restoring mechanism (spring force, gravity component) linear near equilibrium.
Concept / Approach: The defining differential equation is a = d^2x/dt^2 = −ω^2 x, showing acceleration proportional to displacement and directed toward x = 0 (negative sign). Thus, acceleration always points toward the mean position, reversing direction as the particle passes through equilibrium.
Step-by-Step Reasoning:
Assume linear restoring force: F = −k x. Newton’s law: m a = −k x ⇒ a = −(k/m) x. Let ω^2 = k/m ⇒ a = −ω^2 x (toward equilibrium).Verification / Alternative check: Energy view: total energy E = (1/2)k x^2 + (1/2)m v^2 remains constant; slope of potential pulls the mass back toward x = 0, consistent with acceleration direction.
Why Other Options Are Wrong: Saying “False” would contradict the SHM definition; without the “proportional to displacement” part, the motion may not be strictly SHM, but the directionality statement is still correct for SHM.
Common Pitfalls: Forgetting proportionality as well as direction; assuming large-angle pendulum motion is perfectly SHM (it is only approximate).
Final Answer: True.