Hooke’s law
F = k × x
Hooke’s law in the four directions of lab and homework: the force of a stretched spring, its stiffness from a force-extension pair, the extension under a force, and the lab determination par excellence, k = m·g / x with a hanging mass. The elastic energy ½·k·x² is always returned, with the why of the ½: the force grows from zero to k·x, the work is the area of the triangle. The refusals state the physics: a spring of zero stiffness is a string, a zero extension measures nothing.
49.033 N/m
- Calculation
- k = m·g / x = 0.2 kg × 9.80665 / 0.04 m = 49.033 N/m
- Stored elastic energy
- 0.039227 J
The stiffness is 49.033 N/m: at equilibrium, the spring carries exactly the hanging weight.
The stored energy is ½·k·x², and the ½ is no decree: the force grows from zero to k·x during the stretch, the work is the area of the triangle under the line. Quadratic consequence: doubling the extension quadruples the energy.
Stiffness reads in newtons per metre: k = 100 N/m means one newton per centimetre. The larger k, the harder the spring; two identical springs in series are twice as soft, in parallel twice as hard.
The returned force is the one the spring exerts, directed towards its natural length: a restoring force. In equations of motion it is written F = −k·x, the sign only saying it opposes the displacement.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasF = k × x
F = k × x
k = m·g / x (hanging mass at equilibrium)
E = ½ × k × x²
The force is proportional to the extension, and the stiffness k is the slope of that line: newtons per metre. The motionless hanging mass gives k without a dynamometer, because at equilibrium the spring carries exactly the weight. The energy is the area under the force-extension line: a triangle, hence the ½.
- Stiffness k
- · the force per unit extension, in N/m. 100 N/m: one newton per centimetre. It is the spring’s identity card.
- Extension x
- · the offset from the natural length, not the length of the spring. That is the classic lab measurement mistake.
- Restoring force
- · the spring force, always directed towards the natural length. The F = −k·x of equations of motion carries that sign.
- Elastic limit
- · the extension beyond which the spring no longer returns: Hooke’s law stops there, and so does the spring.
Validity domainHooke’s law is the linear regime of a real spring: it holds while the extension stays below the elastic limit, typically a few tens of percent of the natural length for a helical spring.
Hooke’s law is the linear regime of a real spring: it holds while the extension stays below the elastic limit, typically a few tens of percent of the natural length for a helical spring. Beyond, the material deforms permanently and the line ends. Compression follows the same law as long as the coils do not touch. The tool also assumes the ideal spring: massless and without internal friction.
Reading the resultThe equivalent mass translates the force into lab language: 10 N is what one kilogram weighs.
The equivalent mass translates the force into lab language: 10 N is what one kilogram weighs. Combinations follow from the law: in series, springs share the force and add extensions (stiffnesses combine as inverses); in parallel they share the extension and add forces (stiffnesses add). And the quadratic energy explains why a bow is drawn over its whole stroke: the last centimetres store far more than the first.