Theory Exercises

Hooke's Law and Elastic Deformation

Introduction to Deformation

Deformation occurs when a force is applied to an object and changes its shape or size.

There are two types of deformation:

  1. Elastic Deformation: Object returns to original shape after force is removed
- Example: Stretching a spring slightly
  1. Plastic Deformation: Object permanently changes shape after force is removed
- Example: Bending a paperclip

Elastic Force and Restoring Force

When an object is elastically deformed, it develops an elastic force that opposes the deformation and tries to restore the original shape.

This opposing force is called the restoring force.

Characteristic: The restoring force is always directed opposite to the deformation.

Hooke's Law

Definition: The elastic force produced by a deformation is directly proportional to the magnitude of the deformation, provided the elastic limit is not exceeded.

\[F_e = -k \times x\]

Where:
  • F_e = elastic force (Newtons, N)
  • k = spring constant (N/m)
  • x = displacement from equilibrium (meters, m)
  • − = negative sign indicates force opposes displacement

The Spring Constant (k)

The spring constant is a measure of the stiffness of a material or spring.

High k value: Stiffer spring (requires more force to stretch) Low k value: Softer spring (stretches easily) Units: N/m (Newtons per meter)

> [Ejemplo: > - Steel spring: k = 1000 N/m (very stiff) > - Rubber band: k = 50 N/m (easily stretched) > - Foam spring: k = 10 N/m (very soft)]

Understanding Hooke's Law

Linear Relationship

Hooke's Law describes a linear relationship between force and displacement:

  • If you double the displacement → the elastic force doubles
  • If you triple the displacement → the elastic force triples

This linear relationship continues up to the elastic limit.

Elastic Limit

The elastic limit is the maximum deformation a material can undergo while still returning to its original shape.

Beyond the elastic limit:
  • Plastic deformation occurs
  • Permanent damage to the material
  • Hooke's Law no longer applies

Applied Force vs. Elastic Force

When you stretch a spring:

  • Applied force (F_app): The force you exert on the spring
  • Elastic force (F_e): The force the spring exerts back on you

At equilibrium: F_app = F_e

\[F_{app} = k \times x\]

> [Ejemplo: A spring with k = 200 N/m is stretched 0.5 m. > - Applied force needed: F = 200 × 0.5 = 100 N > - Elastic force of spring: 100 N (in opposite direction)]

Applications of Hooke's Law

1. Springs and Springs Systems

Compression Springs

  • Used in mattresses, cushions
  • Resist compression force
  • Push back when compressed

Tension Springs

  • Used in garage doors, springs for hanging weights
  • Resist stretching force
  • Pull back when stretched

Torsion Springs

  • Used in door hinges, clothespins
  • Resist twisting (rotational) force

2. Shock Absorbers and Suspensions

Car Suspensions:
  • Springs absorb impact from bumps
  • Spring constant chosen for comfort vs. stability
  • Dashpots (dampers) prevent oscillation
Building Foundations:
  • Flexible supports reduce earthquake damage
  • Springs and dampers absorb vibrations

3. Materials Testing

Engineers use Hooke's Law to:

  • Determine material properties
  • Test stress-strain relationships
  • Predict material performance
  • Ensure safety in construction

4. Medical Applications

Elastic bandages:
  • Support muscles and joints
  • Allow reasonable compression
  • Spring constant designed for comfort and support
Orthodontic braces:
  • Apply consistent force over time
  • Spring constant adjusted for tooth movement

5. Sports Equipment

Trampoline:
  • Fabric acts like spring (k value specific to material)
  • Stores energy, releases with rebound
Tennis racket strings:
  • Tension creates elastic force
  • Elasticity transfers energy to ball

Elastic Potential Energy

A stretched or compressed spring stores elastic potential energy:

\[E_e = \frac{1}{2}kx^2\]

Where:
  • E_e = elastic potential energy (Joules, J)
  • k = spring constant (N/m)
  • x = displacement (m)

This energy is released when the spring returns to its natural length.

> [Ejemplo: A spring with k = 100 N/m compressed 0.3 m stores: > - E_e = 0.5 × 100 × (0.3)² = 0.5 × 100 × 0.09 = 4.5 J]

Series and Parallel Springs

Springs in Series

When springs are connected end-to-end:

\[\frac{1}{k_{total}} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} ...\]

Result: Total spring constant decreases (system becomes softer)

> [Ejemplo: Two springs with k₁ = 200 N/m and k₂ = 300 N/m in series: > - 1/k_total = 1/200 + 1/300 = 3/600 + 2/600 = 5/600 > - k_total = 600/5 = 120 N/m (softer than either spring alone)]

Springs in Parallel

When springs are side-by-side, sharing the load:

\[k_{total} = k_1 + k_2 + k_3 ...\]

Result: Total spring constant increases (system becomes stiffer)

> [Ejemplo: Two springs with k₁ = 200 N/m and k₂ = 300 N/m in parallel: > - k_total = 200 + 300 = 500 N/m (stiffer than either spring alone)]

Solving Hooke's Law Problems

Step-by-Step Approach

  1. Identify the spring constant (k) or calculate from given information
  2. Determine the displacement (x) from equilibrium position
  3. Apply Hooke's Law: F = k × x
  4. Consider direction: Elastic force opposes displacement

Example Problems

Problem 1: Find Applied Force

Given: Spring constant k = 250 N/m, displacement x = 0.2 m Find: Applied force needed Solution:
\[F = kx = 250 \times 0.2 = 50 \text{ N}\]

Problem 2: Find Spring Constant

Given: Applied force F = 80 N causes displacement x = 0.4 m Find: Spring constant Solution:
\[k = \frac{F}{x} = \frac{80}{0.4} = 200 \text{ N/m}\]

Problem 3: Find Displacement

Given: Spring constant k = 150 N/m, applied force F = 45 N Find: Displacement Solution:
\[x = \frac{F}{k} = \frac{45}{150} = 0.3 \text{ m}\]

Problem 4: Elastic Energy

Given: Spring constant k = 300 N/m, stretched 0.5 m Find: Elastic potential energy Solution:
\[E_e = \frac{1}{2}kx^2 = \frac{1}{2} \times 300 \times (0.5)^2 = 150 \times 0.25 = 37.5 \text{ J}\]

Key Takeaways

  1. Hooke's Law: F_e = -kx (elastic force proportional to displacement)
  2. Spring constant (k): Measure of material stiffness (units: N/m)
  3. Elastic limit: Maximum deformation before permanent damage
  4. Elastic potential energy: E_e = 0.5kx²
  5. Series springs: k_total decreases (softer)
  6. Parallel springs: k_total increases (stiffer)
  7. Applications include: suspensions, shock absorbers, sports equipment, medical devices