Hooke's law calculator for springs

Hooke's law says the force needed to stretch or squash a spring is proportional to how far it is stretched: F = kx, where k is the spring constant in N/m and x is the extension in metres. A spring with k = 250 N/m stretched by 0.08 m needs a force of 250 × 0.08 = 20 N. This Hooke's law calculator finds the force, the spring constant or the extension from the other two, the elastic energy stored, and the combined spring constant of two springs in series or in parallel.

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Hooke's Law Calculator

Find the spring constant, force and extension

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How to use the Hooke's law calculator

1. Open the Hooke's Law Calculator. Under Calculate, choose Hooke's law, Energy or Springs, then what to find.
2. Type the values you know. Pick units under the inputs: force in N or kN, extension in mm, cm or m, and the spring constant in N/m, N/cm, N/mm or kN/m.
3. Read the answer, the working and the energy stored at that extension. The diagram draws the spring at its natural length and stretched.
4. For two springs, choose Series or Parallel, type both spring constants, and add a load if you want the pair's extension.
5. For practice, press Random problem.

Hooke's law and the spring constant

The spring constant measures stiffness: a stiff spring has a big k and stretches little. Hooke's law only holds up to the limit of proportionality. Beyond it the force-extension graph curves, and past the elastic limit the spring no longer returns to its original length.

Here x is the extension in metres and y is the force in newtons. The straight line through the origin is Hooke's law, its gradient is the spring constant, and the area under it is the energy stored. The calculator draws the spring itself rather than this graph, plus an energy against extension curve in its Energy modes.

Elastic potential energy

The energy stored is the area of the triangle under the force-extension line. The elastic potential energy guide covers it in more depth, and the spring constant equation guide covers k.

Worked examples you can check in the calculator

Example 1: the force on a spring

k = 250 N/m, x = 0.08 m. Choose Force: F = 250 × 0.08 = 20 N. The calculator also shows the energy stored, ½ × 250 × 0.08² = 0.8 J.

Example 2: the spring constant, with centimetres

A 6 N weight stretches a spring by 4 cm. Choose Spring constant, set the extension unit to cm and type 6 and 4. The calculator converts 4 cm to 0.04 m: k = 6 ÷ 0.04 = 150 N/m, storing 0.12 J.

Example 3: the extension under a mass

A 1 kg mass hangs on a spring with k = 490 N/m. Its weight is 1 × 9.81 = 9.81 N. Choose Extension: x = 9.81 ÷ 490 = 0.0200 m, or 2.00 cm.

Example 4: energy stored

A stiff spring, k = 800 N/m, is stretched 0.15 m. Choose Energy, then From k and x: E = ½ × 800 × 0.15² = 9 J, with a force of 120 N at that extension. Doubling the extension would store four times the energy, because x is squared.

Example 5: two springs in series and in parallel

Springs of 200 N/m and 300 N/m carry a 12 N load. Joined end to end (Series), 1/k = 1/200 + 1/300, so k = 200 × 300 ÷ 500 = 120 N/m, and the pair stretches 12 ÷ 120 = 0.1 m. Side by side (Parallel), k = 200 + 300 = 500 N/m, and they stretch only 0.024 m. Series springs are softer than either one; parallel springs are stiffer.

Finding k from an experiment

In the classic experiment you hang weights on a spring and measure each extension. Here are some results:

Up to 4 N each newton adds 2.5 cm, so the line is straight and k = 1 ÷ 0.025 = 40 N/m, the gradient of the force-extension graph. At 5 N the extension is 13.5 cm instead of 12.5 cm: the spring has passed its limit of proportionality. The calculator doesn't fit a line to data; take the gradient from your graph, then use k in the calculator.

Common mistakes

• Using the length instead of the extension. x is how much longer the spring got, not its total length.
• Centimetres in F = kx. In N/m, x must be in metres. Set the unit in the calculator, or divide by 100.
• Using mass for force. Multiply kilograms by 9.81 to get the weight in newtons.
• Applying it past the limit of proportionality. F = kx only holds while the graph is straight.
• Mixing up series and parallel. Series: add reciprocals. Parallel: add the constants.

Read next

Physics · MechanicsSpring constant equation: k = F/e and Hooke's lawThe spring constant equation k = F/e from Hooke's law, units, finding k from a force-extension graph, the limit of proportionality, examples and practice.