In classical mechanics and spectroscopy: under what condition will a simple harmonic oscillator (for example, a mass–spring system or a molecular bond modeled as a harmonic oscillator) efficiently absorb energy from an external periodic driving force?

Introduction / Context:
Simple harmonic oscillators are foundational models in physics and physical chemistry, used to describe systems ranging from mass–spring setups to molecular vibrations. A key question is when such an oscillator absorbs energy most efficiently from an external periodic force. This principle underpins resonance in mechanics and selective absorption in vibrational spectroscopy.

Given Data / Assumptions:

• The oscillator has a natural (angular) frequency, often written as omega_0.
• An external driver supplies a sinusoidal force at frequency omega.
• Real systems include some damping, even if small, which enables steady-state energy transfer.

Concept / Approach:
Energy transfer from a driver to an oscillator is governed by resonance. Maximum power absorption occurs when the driver frequency equals the oscillator’s natural frequency (omega = omega_0). Damping broadens the resonance peak but does not change the condition for peak absorption. Off-resonance driving yields far less energy uptake.

Step-by-Step Solution:
Identify the natural frequency of the oscillator: omega_0 = sqrt(k / m) for a mass–spring model.Apply a periodic driving force: F(t) = F0 * sin(omega * t).Steady-state amplitude is largest near omega ≈ omega_0; the average power absorbed from the driver peaks at resonance.Therefore, efficient energy absorption requires frequency matching between the driver and the oscillator.

Verification / Alternative check:
In mechanical demos (e.g., bridge oscillations) and in molecular IR absorption, the strongest response occurs at characteristic frequencies. In spectroscopy, photons at resonant frequencies are absorbed to excite vibrational modes.

Why Other Options Are Wrong:
At any time (a) ignores frequency selectivity. Matching amplitudes (c) is irrelevant; frequency, not amplitude equality, controls resonance. No absorption (d) is false in damped real systems. Twice the frequency (e) corresponds to a higher harmonic, not maximal fundamental absorption.

Common Pitfalls:
Confusing amplitude with frequency; neglecting the role of damping in enabling sustained energy transfer; assuming any periodic force leads to large absorption.

Final Answer:
When the driving and natural frequencies match exactly (resonance).