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The x-t graph of a particle performing simple harmonic motion is shown in the figure. The acceleration of the particle at t = 2 s is:
From graph: T = 8 s ⟹ ω = 2π/8 = π/4 rad/s. At t = 2 s, particle is at positive crest x = +1 m. Acceleration a = -ω² x = -(π/4)²(1) = -π²/16 m s⁻² in 10 seconds!
The given sinusoidal waveform begins at x = 0 at t = 0, reaches its positive maximum x = +1 m at t = 2 s, returns to zero at t = 4 s, reaches its negative minimum x = -1 m at t = 6 s, and completes one full oscillation at t = 8 s:
The angular frequency of the simple harmonic oscillator is related to the time period T by:
Since the motion starts at the origin at t = 0 moving in the positive x direction, the displacement function is represented by:
Differentiating displacement twice with respect to time yields the acceleration function a(t) = -ω² x(t). At time t = 2 s, the displacement is x(2) = sin(π/2) = +1 m:
Diagnose every choice: Know exactly why the right option works and why distractors fail.
Accurate acceleration at positive extreme: a = -ω² A = -π²/16 m s⁻².
Wrong denominator and omitted restoring negative sign.
Incorrect denominator (calculated 4² = 8 instead of 16).
Positive sign error.
At the positive displacement extreme (x = +A), the restoring force and acceleration act BACKWARDS toward the mean position, meaning acceleration MUST be negative (a = -ω² A = -π²/16). Forgetting the minus sign leads to Option (4).
"In SHM, acceleration is directly proportional to displacement and directed opposite to it: a = -ω² x."
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