Hooke’s Law and Spring Experiment — Procedure, Calculations and Verification for B.Tech

The Hooke’s Law experiment is one of the most fundamental experiments in the Strength of Materials and Engineering Mechanics laboratory. It forms the experimental foundation for understanding elastic behaviour, spring stiffness, and the elastic limit of materials — concepts that underpin the design of springs, shock absorbers, seals, and structural members.

Aim of the Experiment

1. To verify Hooke’s Law: that the extension of a spring is directly proportional to the applied load within the elastic limit.
2. To determine the stiffness (spring constant) k of the given spring.
3. To find the elastic limit of the spring.

Theory

Hooke’s Law

Hooke’s Law states: Within the elastic limit, the deformation (extension or compression) of a spring is directly proportional to the applied force.

F = k × x

where F = applied force (N), k = spring stiffness or spring constant (N/m or N/mm), x = extension or compression (m or mm)

This law was formulated by Robert Hooke in 1678 and applies to all elastic materials within their proportional limit — not just springs but also beams, columns, shafts, and rubber components.

Spring Constant (Stiffness)

The spring constant k represents the force required per unit deflection:

k = F/x = W/δ (N/m)

The stiffness depends on the spring material, wire diameter, coil diameter, and number of active coils:

k = Gd⁴ / (8D³n)

where G = shear modulus of spring material, d = wire diameter, D = mean coil diameter, n = number of active coils

Types of Springs

• Close-coiled helical spring: Used in compression and extension. Most common type in engineering.
• Open-coiled helical spring: Used where both compression and torsion must be accommodated.
• Leaf spring: Used in vehicle suspensions. Multiple flat strips act together.
• Torsion spring: Stores energy in torsion rather than bending or compression.
• Disc spring (Belleville): Very stiff, used in high-load applications with small deflection.

Elastic Limit vs. Proportional Limit

Within the elastic limit, removing the load returns the spring to its original length (no permanent set). Beyond the elastic limit, the spring takes a permanent set. The proportional limit is where Hooke’s Law ceases to hold (F–x graph deviates from a straight line). In practice, for springs, the proportional limit ≈ elastic limit.

Apparatus Required

• Spring testing apparatus (vertical frame with graduated scale)
• Helical extension spring or compression spring (one or more of different stiffness)
• Weights and weight hanger (50 g to 2 kg)
• Vernier caliper (for measuring wire diameter d and coil diameter D)
• Steel rule (for measuring spring free length and counting coils)
• Dial gauge (0.01 mm resolution) — alternative to scale reading for precision

Procedure

1. Measure the free (unloaded) length of the spring (L₀). Record wire diameter d, mean coil diameter D, and count active coils n. Record material type if known.
2. Mount the spring on the testing apparatus. Attach the weight hanger without any additional weight. Note the initial reading on the graduated scale (or zero the dial gauge).
3. Add the first weight (e.g., 100 g = 0.981 N). Record the new scale reading. Calculate extension x = new reading – initial reading.
4. Continue adding weights in equal increments. Record extension for each load.
5. After reaching maximum safe load, remove weights one at a time and record the reading at each load during unloading.
6. Check whether loading and unloading readings match (if they match, the spring is within its elastic limit).
7. Plot F vs. x (load–extension graph). The slope gives k.

Observation Table

Calculations

From the load-extension graph (straight line portion): slope = k = ΔF/Δx (N/mm)

Theoretical spring constant: k_th = Gd⁴/(8D³n)

For steel spring: G = 80 GPa = 80,000 N/mm²

Percentage error = (k_actual – k_theoretical)/k_theoretical × 100%

Sample Calculation

d = 3 mm, D = 30 mm, n = 10 active coils, G = 80,000 N/mm²

k_th = 80000 × 3⁴ / (8 × 30³ × 10) = 80000 × 81 / (8 × 27000 × 10) = 6,480,000 / 2,160,000 = 3.0 N/mm

Result

Experimental spring constant k = _____ N/mm
Theoretical spring constant k_th = _____ N/mm
Percentage error = _____%
Hooke’s Law is verified: extension is proportional to load within the elastic limit (up to _____ N applied load).

Springs in Series and Parallel

Series combination: 1/k_eq = 1/k₁ + 1/k₂ + 1/k₃ (more flexible — lower stiffness)

Parallel combination: k_eq = k₁ + k₂ + k₃ (stiffer — higher stiffness)

Viva Questions

1. State Hooke’s Law and give its limitations.
2. What is spring stiffness? How does it depend on wire diameter and number of coils?
3. What happens to a spring when loaded beyond its elastic limit?
4. How do springs in series differ from springs in parallel in terms of stiffness?
5. What is the shear modulus of elasticity and how is it related to spring design?
6. Give three engineering applications of helical springs.
7. What is a progressive spring? Where is it used?

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