Hooke’s Law Formula

Hooke’s Law Formula (F = −kx)

The Hooke’s Law formula is a foundational principle in mechanics that explains the behavior of elastic materials when subjected to a deforming force. It is a vital concept in the CBSE Class 11 Physics syllabus (Mechanical Properties of Solids) and is heavily tested in engineering and medical entrance exams like JEE and NEET.

Class: 11
Topic: Elasticity & Solids
Exams: CBSE · JEE · NEET

What is Hooke’s Law?

Hooke's Law states that the displacement or size of the deformation of an elastic object is directly proportional to the deforming force or load acting upon it. Crucially, this law only holds true within the elastic limit of the material. Once a material crosses this threshold, it permanently deforms or breaks.

✓ Key Takeaway
In short, if you pull a spring twice as hard, it will stretch exactly twice as much—provided you do not stretch it so much that it loses its ability to bounce back.

1. Hooke’s Law Formula for Springs

When applied to a linear spring system, the relationship between the force applied and the extension or compression can be written mathematically as:

Spring Equation
F = −k × x

Where:

  • F = Restoring force exerted by the spring (in Newtons, N)
  • k = Spring constant or stiffness factor (in N/m)
  • x = Displacement, extension, or compression from the mean/equilibrium position (in meters, m)

Note: The negative sign indicates that the restoring force exerted by the spring acts in the direction exactly opposite to the displacement. If you pull the spring down, the restoring force pulls it back up. If a question asks purely for the magnitude of the external force required to stretch it, the equation is written as F = kx.

2. General Form of Hooke’s Law (Stress & Strain)

For solid materials like metal wires, beams, and rods, Hooke's Law is expressed in terms of internal restoring forces and relative deformations. Within the elastic limit, Stress is directly proportional to Strain.

Stress-Strain Relationship
Stress ∝ Strain  ⇒  Stress = E × Strain

Where E is the Modulus of Elasticity of the material. Depending on the type of load applied, this constant can take different forms:

  • Young's Modulus (Y): For linear tensile or compressive stress/strain (stretching a wire).
  • Bulk Modulus (B): For volume stress/strain (compressing an object uniformly in a fluid).
  • Shear Modulus / Modulus of Rigidity (η): For tangential or shearing stress/strain.

The Stress-Strain Curve & Hooke’s Law Limit

Hooke's Law does not apply to the entire lifespan of a solid material. On a standard stress-strain graph, Hooke’s Law is perfectly valid only along the initial straight-line segment from the origin to the Proportionality Limit.

  • Proportionality Limit: The exact point up to which stress is directly proportional to strain (straight line path).
  • Elastic Limit (Yield Point): The absolute limit up to which the material behaves elastically. Beyond this, if you remove the force, the material stays permanently stretched.

Units and Dimensions of the Spring Constant (k)

When applying the spring formula, you will frequently need to verify units for dimensional consistency:

  • S.I. Unit of Spring Constant (k): Newton per meter (N/m) or kg/s2
  • C.G.S Unit of Spring Constant: Dyne/cm
  • Dimensional Formula: [M1 L0 T-2]

Common Mistakes to Avoid in Numericals

× Using Extensions in Centimeters (cm)
Physics problems often mention a spring stretching by "5 cm". Do not plug "5" into F = kx! Always convert centimeters to meters by multiplying by 10-2 (or dividing by 100) first.
× Mixing Total Length with Extension
If a spring has an original length of 10 cm and stretches to a final length of 14 cm when a weight is hung, the value of x is 4 cm (0.04 m), not 14 cm. Displacement is always the change in length.

Solved Numerical Examples on Hooke’s Law

Example 1: Finding Spring Restoring Force

A spring with a spring constant of 250 N/m is compressed by 8 cm from its equilibrium position. Calculate the magnitude of the force acting on the spring.

Given:
Spring constant (k) = 250 N/m
Compression (x) = 8 cm = 8 × 10-2 m = 0.08 m

Calculation:
Using the magnitude form of Hooke’s Law: F = k × x
F = 250 × 0.08
F = 20 N
Answer: Force = 20 N

Example 2: Hanging Mass Problem (JEE/NEET Level)

A mass of 3 kg is suspended from a vertical spring hanging from a ceiling. If the spring stretches by 6 cm, determine its stiffness constant. (Take g = 10 m/s2)

Given:
Mass (m) = 3 kg ⇒ Downward Deforming Force (F) = m × g = 3 × 10 = 30 N
Extension (x) = 6 cm = 0.06 m

Calculation:
According to Hooke’s Law: F = kx ⇒ k = F / x
k = 30 / 0.06
k = 3000 / 6 = 500
Answer: Spring Constant (k) = 500 N/m

Frequently Asked Questions (FAQs)

What is the primary formula for Hooke’s Law?
The structural formula for a spring system is F = −kx, where F is the restoring force, k is the spring constant, and x is the displacement. The negative sign represents that the force works opposite to the direction of extension.
What does the spring constant (k) represent?
The spring constant (k) is a measure of the stiffness of the spring. A high value of k means the spring is stiff and require a lot of force to pull or compress, whereas a small k value means it stretches easily.
How is Hooke’s Law expressed for solid engineering materials?
For solid rods, wires, and materials, it is expressed as Stress = E × Strain, where E is the Modulus of Elasticity of that specific material.
Does Hooke's Law work under all conditions?
No, Hooke's Law holds true only as long as the material stays within its elastic limit (proportionality limit). If a material is overstretched beyond this range, it will undergo plastic, permanent deformation.