Simple harmonic motion (SHM) – relation between acceleration and displacement\nFor a particle executing SHM, the instantaneous acceleration a in terms of angular frequency ω and displacement y from the mean position is:

Introduction / Context:
In simple harmonic motion (SHM), restoring forces lead to an acceleration proportional to displacement and directed toward the mean position. This item checks the fundamental proportionality between acceleration a, angular frequency ω, and displacement y.

Given Data / Assumptions:

• Displacement from mean position = y.
• Angular frequency of oscillation = ω (in rad/s).
• Motion is ideal SHM (no damping, no driving force).

Concept / Approach:
The defining differential equation of SHM is a = d^2y/dt^2 = −ω^2 y. The magnitude of acceleration is ω^2 |y| and its direction is opposite to the displacement, producing the restoring tendency.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional analysis: [ω^2 y] has dimensions of acceleration since [ω] = s^−1 and [y] = m. Hence [ω^2 y] = m s^−2, consistent with acceleration.

Why Other Options Are Wrong:

• ω · y: missing a factor of ω; dimensions are m s^−1, which is velocity, not acceleration.
• ω^2 / y: yields dimensions s^−2 m^−1, not acceleration.
• ω^3 · y: incorrect power of ω; dimensions m s^−3.

Common Pitfalls:
Ignoring the negative sign (direction). In many MCQs, the magnitude is asked implicitly; always remember the acceleration acts toward the mean position.

Final Answer:
a = −ω^2 y (magnitude ω^2 · y, opposite to displacement)