The Thinking Experiment

Hooke's Law Simulation

Investigate the relationship between the force applied to a spring and how much it stretches. Build a model that describes this relationship.

Learning Objective

Collect force-stretch data from a virtual spring, graph the results, and determine the spring constant k. You will not be told what equation to use — you will discover it.

Step 1 · Set Up

The position is zero when nothing is hanging (natural length). The 50 g hanger is already attached and causing an initial stretch (Δx). Add mass in 50 g increments.

Step 2 · Record

After each mass addition, wait for the spring to settle, then click Record Data Point to capture the trial into your data table.

Step 3 · Analyze

Watch your Force vs. Δx graph build in real time. Toggle the best-fit line to discover the slope — and the spring constant.

Spring Setup

Add masses and record data points to build your graph.

Hanging Mass0.050 kg
Weight (Force)0.000 N
Stretch (Δx)0.0000 m
Trials Recorded0 / 8

Controls

0 g on hanger

Data Table

# Added (g) Total (kg) Δx (m) Force (N)
1————
2————
3————
4————
5————
6————
7————
8————

Force vs. Δx

🔒 Teacher Mode

⚠ Students: do not open — discover k from your data!

🔮 Predict First

If you double the mass on the spring, what happens to the stretch (Δx)? Less, same, more, or exactly twice as much? Make a prediction before collecting data!

📊 Analyze the Graph

What shape does the Force vs. Δx graph make? Toggle the best-fit line. The slope has a physical meaning — "for every 1 m of stretch, the spring needs ___ N."

🧪 Build Your Model

Write an equation: F = ___. The constant k is the spring constant — it tells you how stiff the spring is. Test it with a prediction!

The Model

Gravity pulls the hanging mass down with force \(F = mg\). At equilibrium the spring force balances gravity:

\(F_s = k\Delta x\)

where k is the spring constant (N/m) and Δx is the stretch (m).

What Does the Slope Tell You?

Plotting Force (y) vs. Stretch (Δx) gives a straight line. The slope equals k:

\(\text{slope} = \dfrac{\Delta F}{\Delta x} = k\)

Key Insights

  • Linear: Double the force → double the stretch (Δx).
  • Larger k = stiffer spring.
  • Elastic limit: Hooke's Law only holds if the spring returns to its natural length.

When Does It Break Down?

Beyond the elastic limit the spring permanently deforms and the linear relationship fails. This is why the lab says: "Stop if the spring does not return to its original length."

Learning Objective

Explore Simple Harmonic Motion — adjust the spring constant, mass, and damping to see how they affect oscillation, period, and energy exchange.

Spring Oscillation

Displacement (Δx)1.00 m
Velocity0.00 m/s
Max Vel0.00 m/s
Min Vel0.00 m/s
Spring Force-30.00 N
Period1.62 s
Time0.00 s

Real-Time Graphs

Δx vs Time

Velocity vs Time

Controls

🔧 Spring Stiffness

Keep mass at 2 kg. Increase k from 10→80. A stiffer spring oscillates faster.

⚖ Mass Effect

Keep k = 30. Try mass 1→8 kg. More mass = slower oscillation. Period increases.

💨 Damping

Increase damping from 0→3 to see amplitude decay. Real springs always have some friction.

Simple Harmonic Motion

\(F=-kx \;\Rightarrow\; a=-\frac{k}{m}x\)

Period

\(T=2\pi\sqrt{\frac{m}{k}}\)

Period depends only on mass and spring constant — not amplitude.

Energy

\(PE=\tfrac{1}{2}kx^2 \qquad KE=\tfrac{1}{2}mv^2\)

Total mechanical energy is conserved in an ideal (undamped) system.

⚡ Energy Inquiry

Discover how elastic potential energy depends on displacement — and verify that mechanical energy is conserved. You will build the formula PE = ½kΔx² from your own data.

Spring Oscillation

Start the sim, pause at different positions, and capture snapshots.

Displacement (Δx)0.00 m
PE (½kΔx²)0.00 J
KE (½mv²)0.00 J
Total E0.00 J

Controls

Keep Damping at 0 for Parts 1–3. Use damping only for the extension in Part 4.

🔮 Part 1 · Predict

Before collecting data, answer these in your notebook:

  1. If you stretch the spring twice as far, does it store twice the energy, less than twice, or more than twice? Why?
  2. At the equilibrium position (Δx = 0), where is all the energy? What about at the maximum stretch?
  3. Do you expect the total energy to stay constant or change over time (with no damping)?

📊 Part 2 · Collect Data

Set damping to 0.0. Click Start, then Pause at several different positions. Press ⚡ Capture Snapshot above to record. Aim for at least 6 snapshots spread across a range of Δx values.

#Δx (m)Δx² (m²)PE (J)KE (J)Total E (J)
No data yet — capture snapshots above.

📈 Part 3 · Graph & Discover

The graph plots your PE (y-axis) vs. Δx² (x-axis). Toggle the best-fit line and read the slope.

💡 What does the slope equal? How does it relate to k?

🧠 Part 4 · Reflect & Conclude

  1. Look at the Total E column. Is it constant? What does this tell you about energy in the spring system?
  2. Your PE vs. Δx² graph should be a straight line through the origin. What relationship does this reveal between PE and Δx?
  3. The slope equals ½k. Write the complete formula: PE = ½kΔx² — confirm your slope matches ½k from your settings.
  4. When KE = 0, where is the mass? Write an equation relating PEmax to total energy using PE = ½kΔx².
  5. Extension: Enable damping > 0 and re-run. What happens to Total E over time? Where does the energy go?