Is it SHM? Plot a against x.

Acceleration-vs-displacement plot for several periodic motions (spring, bouncing ball, sin² ωt, marble in a bowl): only a straight line through the origin with negative slope is SHM.

Find the amplitude at which the bowl marble's a-x plot stops being a straight line (≥ 1 % off).

Every periodic motion is SHM.

• Newton's second law with a = −ω²x gives the force on a particle of mass m in SHM: F = ma = −mω²x, written F = −kx with k = mω².
• So ω = √(k/m) and the period is T = 2π√(m/k). A stiffer spring makes it faster; a heavier block makes it slower.
• The force always points towards the mean position, which is why it is called the restoring force.
• SHM can be defined in either of two equivalent ways: by the displacement x(t) = A cos(ωt + φ), or by the force law F = −kx. Differentiating the first twice gives the second; integrating the second twice gives back the first.
• A body acted on by a force exactly proportional to x is a linear harmonic oscillator. If the force also contains small terms in x², x³ and so on, it is a non-linear oscillator.
• Test for SHM: the acceleration must be a negative constant times x. a = −(constant) × x is SHM; an acceleration proportional to +x, to x² or to x³ is not.
• Worked example: a block of mass m held between two identical springs of constant k fixed on either side. Displaced by x, one spring stretches and the other compresses by x, and both push it back, so F = −2kx and T = 2π√(m/2k).
• The period of SHM does not depend on the amplitude, the energy or the phase constant; it is set only by m and k.

Teaching this idea? Put the simulation on your own page or LMS with this code.