Elasticity · One dimension
Springs and Hooke’s law
Why do two springs stretch by different amounts under the same force?
Explore force, deformation and stiffness with two springs. Release them to see how inertia and damping shape their return to equilibrium.
Try the interactive demoStart with something familiar
Pull gently on two different springs. One stretches more under the same pull. Stiffness names this resistance to deformation; it does not mean that the spring is stronger against breaking.
k = 100 N/m
Extension 60 mm · F = 6 N
k = 200 N/m
Extension 30 mm · F = 6 N
Notice what changes
First compare at equal force. The spring that extends less has greater stiffness. Comparing different forces would mix up the effect of the load with the effect of the spring.
Understand why
In the proportional range, twice the pull produces twice the extension. The hand must keep pulling to hold it there; the spring pulls back in the opposite direction.
Connect it to wood and making
A wood strip also resists deformation, but its size matters. A thicker strip can feel stiffer without being made of wood with a larger modulus. The spring analogy introduces stiffness; modulus separates out the geometry.
Put names to the ideas
Force and deformation
Within the linear elastic range, the force required to hold a spring is proportional to its extension. Its restoring force points back toward equilibrium.
Stiffness and modulus
Spring stiffness k describes the response of an object. Material modulus E describes a material property; the stiffness of a plate also depends on its thickness, arching and shape.
What happens on release?
The applied force disappears on release. The restoring force and inertia produce oscillation; damping dissipates energy until the motion settles.
F is the holding force, x is extension, and k is the force needed per unit extension. The minus sign in restoring force only indicates that the spring pulls toward its original length.
A worked example
A 6 N holding force stretches a 100 N/m spring by 60 mm and a 200 N/m spring by 30 mm. Doubling stiffness halves extension under the same force; releasing the spring is a different, dynamic experiment.
Explore it yourself
Apply the same 6 N force to both springs, then change their stiffness ratio. Release them and compare their motion; the 6 N markers and peak records preserve the stretched distances.
HOOKE’S LAW / ONE DIMENSION
Tension and compression
Drag to load. Release to oscillate.
- Extension at 6 N
- 60.0 mm
- Peak deformation
- 0.0 mm
- Extension at 6 N
- 30.0 mm
- Peak deformation
- 0.0 mm
Static equilibrium
- F
- Applied force
- k
- Stiffness
- Δx
- Deformation