Preview
Hooke's Law
Restoring force, oscillation period, and energy — F = −kx in motion
In 1660 Robert Hooke published his greatest discovery as a scrambled anagram — ceiiinosssttuv — too valuable to share, too precious to lose. Unscrambled: ut tensio, sic vis. As the extension, so the force. Stretch a spring twice as far and it pulls back exactly twice as hard, always aimed home. That one proportionality, F = −kx, is why a displaced mass doesn't just sag or fly away — it oscillates, endlessly trading kinetic energy for spring energy and back. Now load the spring heavier and ask your gut: does the clock tick faster or slower? Most guts say faster — more weight, more force, more action. But the force at a given stretch hasn't changed at all; F = −kx has no m in it. What changed is the inertia the spring has to turn around: T = 2π√(m/k), so quadruple the load and the period doubles. It's why a loaded car sits lower and rides slower on its suspension, and why atomic force microscopes can weigh molecules with a vibrating microcantilever. In this lab you'll load the spring from 0.1 to 5 kg and clock it, swap soft for stiff springs across a 25-fold range, flip the release into compression and watch the force flip sign with it, then switch on damping and watch the energy books drain into heat — the one thing an ideal oscillator never does.
What you'll be able to do
- Read the restoring force F = −kx as a sign-and-magnitude statement: released at x₀ = +0.30 m the panel shows −24.00 N (pulling back up), released at −0.30 m it flips to +24.00 N (pushing back down) — always aimed at the natural length, with the k slider scaling it live
- Apply T = 2π√(m/k) across both levers: quadrupling the mass 0.5 → 2.0 kg doubles the period 0.497 → 0.993 s (√4 = 2), while the full k lever from Soft (20 N/m, 0.993 s) to Stiff (300 N/m, 0.257 s) — 15× stiffer — buys only √15 ≈ 3.9× on the clock; confirm each against the zero-cross T (meas) row
- Track the energy books both ways: undamped, E = ½mv² + ½kx² holds the release value 3.60 J at defaults through every KE↔PE trade; with the Damped preset (c = 0.3 N·s/m) the drag force −cv drains ~26% of the total per cycle into heat — envelope e^(−ct/m) on the energy, 90% gone within ~3.8 s — while T (theory) never leaves 0.497 s
Formulas
Make a prediction
Default state: k = 80 N/m, m = 0.5 kg, released from rest at x₀ = 0.30 m — T (theory) reads 0.497 s and the release force is −24.00 N. Now quadruple the mass to 2.0 kg (same spring, same release point) and hit Reset Sim. What does the period do?
No grading here — commit to a guess, then scroll down and test it yourself.
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Your prediction
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Answer: T doubles to 0.993 s — the spring's pull at each stretch is unchanged, but there is four times the inertia to turn around: T = 2π√(m/k), and √4 = 2
T = 2π√(m/k) = 2π√(2.0/80) = 0.993 s — exactly double the default 2π√(0.5/80) = 0.497 s. The force half of the intuition is real: at 0.30 m the spring pulls with 24 N whatever is hanging on it, because F = −kx contains k and x and nothing else. But force is only half of Newton's second law — a = F/m, so quadrupling m quarters every acceleration in the cycle, and the clock stretches by √4 = 2. Option B is the sim's own flagged misconception ('Students often think heavier masses oscillate faster. Wrong — larger m means larger inertia'). Option C borrows the pendulum's famous mass-independence — true for pendulums, where the restoring force is gravity and gravity itself scales with m, cancelling it; false for springs, whose restoring force couldn't care less what's attached. That contrast is this batch's twin lab, Pendulum Lab — run them back to back.
Quiz (0/3)
A 0.5kg mass stretches a spring by 5cm. What is the spring constant?
How much elastic PE is stored when a spring (k=50 N/m) is stretched 10cm?
With the Damped preset the Total energy bar falls steadily. Is energy being destroyed?
You can now
- Read the restoring force F = −kx as a sign-and-magnitude statement: released at x₀ = +0.30 m the panel shows −24.00 N (pulling back up), released at −0.30 m it flips to +24.00 N (pushing back down) — always aimed at the natural length, with the k slider scaling it live
- Apply T = 2π√(m/k) across both levers: quadrupling the mass 0.5 → 2.0 kg doubles the period 0.497 → 0.993 s (√4 = 2), while the full k lever from Soft (20 N/m, 0.993 s) to Stiff (300 N/m, 0.257 s) — 15× stiffer — buys only √15 ≈ 3.9× on the clock; confirm each against the zero-cross T (meas) row
- Track the energy books both ways: undamped, E = ½mv² + ½kx² holds the release value 3.60 J at defaults through every KE↔PE trade; with the Damped preset (c = 0.3 N·s/m) the drag force −cv drains ~26% of the total per cycle into heat — envelope e^(−ct/m) on the energy, 90% gone within ~3.8 s — while T (theory) never leaves 0.497 s