Read a complete oscillation in three motion graphs

Follow time, not just position

The earlier spring-energy investigation selected a position on one half-cycle. This lab selects elapsed time after release, so you can inspect both directions of travel. A 1 kg block starts at x = +0.200 m with zero velocity. The fixed spring stiffness is k = π² N/m, approximately 9.870 N/m, chosen to give a convenient 2 s period for this mass.

Before moving the time control, predict the first equilibrium crossing. The spring pulls toward equilibrium, so the block begins moving left. As it approaches x = 0, the restoring force weakens, but the accumulated velocity does not vanish. At 0.5 s, displacement and acceleration are zero while velocity is approximately −0.628319 m/s.

Compare the three graphs at one instant

The plots share the same 0–4 s time axis and mark the same selected instant. Their vertical axes have different units and fixed ranges: displacement in metres, velocity in metres per second, acceleration in metres per second squared. Equal visual heights across different plots do not mean equal physical quantities.

At 0 s, x = +0.200 m, v = 0 and a ≈ −1.973921 m/s². At 0.5 s, x = 0, v ≈ −0.628319 m/s and a = 0. At 1 s, x = −0.200 m, v = 0 and a ≈ +1.973921 m/s². At 1.5 s, x = 0 again, but v ≈ +0.628319 m/s. At 2 s, the complete starting state returns.

Capture 0.5 s and 1.5 s. The block has the same position, restoring force and energy at both crossings, but opposite velocity. Position alone does not specify where it will move next. The two turning points likewise have zero velocity but nonzero acceleration, which reverses the motion.

Relate slopes, force and phase

Velocity is the slope of the displacement-time graph. At a displacement maximum or minimum, that slope is zero. Near an equilibrium crossing, displacement changes most rapidly, so the velocity magnitude is largest. Acceleration is the slope of the velocity-time graph; velocity’s extrema occur where acceleration is zero.

The force law is F = −kx, where F is signed restoring force, k is positive stiffness and x is displacement from equilibrium. Newton’s law then gives a = −(k/m)x, with mass m in kilograms. Acceleration always points opposite displacement. It does not always point opposite velocity: when the block travels toward equilibrium it speeds up, and when it travels away it slows down.

Phase describes where the motion is in its repeating cycle. The velocity curve is shifted by a quarter-cycle relative to displacement; acceleration has the opposite sign to displacement at every instant. Use the explicit states and slopes to interpret those relationships rather than memorizing a shift direction without a sign convention.

A compact formula for this release

Let omega = √(k/m), measured in radians per second. At elapsed time t, measured in seconds, the phase angle is omega t. With amplitude A = 0.200 m and release from rest at +A:

x = A cos(omega t), v = −A omega sin(omega t), and a = −omega² x.

Cosine and sine describe the repeating phase. At phase 0, π/2, π, 3π/2 and 2π, the listed quarter-cycle states follow. A different starting position or velocity would require a different phase or amplitude. The OpenStax simple-harmonic-motion reference gives further background on these relations.

Optional: verify the formulas through rates of change

Differentiating x = A cos(omega t) gives v = −A omega sin(omega t); differentiating again gives a = −A omega² cos(omega t) = −omega² x. With omega² = k/m, this satisfies ma = −kx and the specified release conditions. The lab evaluates these formulas directly rather than advancing a numerical time integrator.

Transfer without calculating a sine

On the return half-cycle, the block is at x = +0.100 m and moving right. Give the signs of force and acceleration, and decide whether its speed is increasing or decreasing. Could the same position occur while speeding up?

Check the harmonic-motion transfer

Force and acceleration are negative because x is positive. With positive velocity and negative acceleration, speed is decreasing. On the first half-cycle at the same positive position, velocity is negative, so negative acceleration increases speed. The position is the same; direction of travel changes the speed trend.