Simple harmonic motion is defined by an acceleration that is always

Proportional to displacement and directed opposite to it. In SHM, acceleration is proportional to displacement and directed toward equilibrium. The relationship is a = −ω²x, with constant positive ω². Acceleration is zero at equilibrium, not throughout the motion. It is not generally proportional to velocity.

Where is acceleration magnitude greatest in ideal linear SHM?

At both turning points. Acceleration magnitude equals ω²|x|. Displacement magnitude is greatest at both turning points. At equilibrium, acceleration is zero and speed is maximum. Signed accelerations at the two turning points have opposite signs.

An oscillator completes 20 cycles in 10 seconds. What is its period?

0.5 s. Frequency is 20/10 = 2 Hz. Period is the reciprocal, so T = 1/2 = 0.5 s. Two cycles per second does not mean two seconds per cycle.

A simple pendulum keeps small swing angles and unchanged local gravity. What happens when its length becomes four times larger?

Its period doubles. The small-angle period is proportional to √L at fixed g. A fourfold length gives √4 = 2 times the period. Mass does not appear in this ideal formula.

An ideal horizontal spring oscillator doubles its amplitude without changing mass or stiffness. What happens to total mechanical energy?

It becomes four times larger. Total energy is E = ½kA². Doubling amplitude multiplies A² by four, so energy quadruples. The same ideal oscillator's period remains unchanged.

At x = A/2, what fraction of total energy is kinetic in an ideal horizontal spring oscillator?

Three-quarters. Potential energy is proportional to x². At half the amplitude, U = E/4. Conservation of mechanical energy therefore gives K = E − E/4 = 3E/4.

Which description best fits resonance in a lightly damped, driven oscillator?

Driving near a natural frequency can produce a large response. A driving frequency near a natural frequency can produce a strong response. Damping limits that response and affects the resonance peak. Real oscillators do not gain unlimited amplitude merely because they are driven near resonance.

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The mass on an ideal spring becomes four times larger while stiffness remains unchanged. How does the period change?

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