Identify the incorrect statement for simple harmonic motion (SHM) Which of the following statements about SHM is incorrect?

Introduction / Context:
In simple harmonic motion (SHM), displacement x, velocity v, and acceleration a have well-defined phase relations. Knowing where these kinematic quantities are maximum or zero is essential in vibrations and oscillations analysis.

Given Data / Assumptions:

• Standard SHM with a = −ω^2 x.
• Motion is about a mean (equilibrium) position.
• No damping or external forcing.

Concept / Approach:

From a = −ω^2 x, acceleration magnitude is proportional to |x|. Thus |a| is maximum at extreme displacements and zero at the mean position. Velocity is the time derivative of displacement and reaches its maximum magnitude at the mean position, becoming zero at the ends of stroke.

Step-by-Step Solution:

Verification / Alternative check:

Using energy: total energy E = (1/2) k A^2. At ends, all energy is potential ⇒ v = 0; restoring force kx is maximum ⇒ acceleration is maximum, consistent with the differential equation.

Why Other Options Are Wrong:

(a), (b), and (d) are correct properties of SHM. (e) is also correct since a = −ω^2 x indicates opposite signs of a and x.

Common Pitfalls:

Confusing velocity and acceleration maxima; thinking acceleration is tied to speed rather than displacement in SHM.

Final Answer:

Acceleration is minimum at the ends of stroke