Spring Force Formula

The Formula

F = -kx (spring constant times displacement)

When to use: Stretch a spring twice as far, it pulls back with exactly twice as much force.

Quick Example

A spring scale: hang a 2kg mass, spring stretches twice as much as for 1kg.

Notation

F is the restoring force in newtons (N), k is the spring constant in N/m (a measure of stiffness), and x is the displacement from the equilibrium position in metres. The negative sign indicates the force opposes the displacement.

What This Formula Means

The restoring force exerted by a spring, proportional to how much it's stretched or compressed.

Stretch a spring twice as far, it pulls back with exactly twice as much force.

Formal View

Hooke's law states that the restoring force of an ideal spring is F = -kx, where the force is linearly proportional to displacement and directed opposite to it. This holds for small deformations within the elastic limit.

Worked Examples

Example 1

easy
A spring with spring constant k = 150 \text{ N/m} is stretched 0.2 \text{ m} from its natural length. What is the restoring force?

Solution

  1. 1
    Apply Hooke's law: F = -kx, where x is the displacement from equilibrium.
  2. 2
    |F| = kx = 150 \times 0.2 = 30 \text{ N}
  3. 3
    The negative sign indicates the force is directed opposite to the displacement (restoring force).

Answer

F = 30 \text{ N toward equilibrium}
Hooke's law states that the restoring force of a spring is proportional to its displacement from the natural length. The force always acts to return the spring to its equilibrium position.

Example 2

medium
A spring stretches 0.04 \text{ m} when a 2 \text{ kg} mass is hung from it. What is the spring constant? How much will it stretch with a 5 \text{ kg} mass? Use g = 9.8 \text{ m/s}^2.

Example 3

medium
A spring with k = 200 \text{ N/m} is compressed by 0.15 \text{ m}. Find the spring force and the elastic potential energy stored.

Common Mistakes

  • Measuring displacement from the wrong reference point — x must be measured from the spring's natural (relaxed) length, not from some other position.
  • Ignoring the negative sign and getting the force direction wrong — the restoring force always opposes the displacement.
  • Applying Hooke's law beyond the elastic limit where the spring deforms permanently and the linear relationship F = -kx no longer holds.

Why This Formula Matters

Model for many oscillating systems; basis of simple harmonic motion.

Frequently Asked Questions

What is the Spring Force formula?

The restoring force exerted by a spring, proportional to how much it's stretched or compressed.

How do you use the Spring Force formula?

Stretch a spring twice as far, it pulls back with exactly twice as much force.

What do the symbols mean in the Spring Force formula?

F is the restoring force in newtons (N), k is the spring constant in N/m (a measure of stiffness), and x is the displacement from the equilibrium position in metres. The negative sign indicates the force opposes the displacement.

Why is the Spring Force formula important in Physics?

Model for many oscillating systems; basis of simple harmonic motion.

What do students get wrong about Spring Force?

Only valid within the elastic limit—stretch too far and the spring deforms permanently.

What should I learn before the Spring Force formula?

Before studying the Spring Force formula, you should understand: force.

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