Elasticity

Elasticity Notes

Overview

• Elasticity: Property of a body to regain its original shape and size when the deforming force is removed.

• Deforming Force: A force that changes the shape and size of the body, originating from outside the body, directed away from equilibrium, and resulting in stress or strain.

• Restoring Force: Force developed inside the body that tries to bring the body back to its initial size and shape, directed towards equilibrium.

Forces

Deforming Force

• Definition: A force that changes the shape and size of an object (page 112).

• Origin: From outside the body.

• Effect: Causes a change in shape.

• Direction: Away from equilibrium.

• Results: Leads to stress or strain.

• Examples:

  • Tension (stretching)

  • Compression

Restoring Force

• Definition: A force that develops inside the body and tries to return the body to its initial size and shape (page 113).

• Origin: From inside the body.

• Effect: Returns the object to its original shape.

• Direction: Towards equilibrium.

Deforming vs. Restoring Forces

• Equal in magnitude but opposite in direction (Newton’s 3rd Law).

• Force diagram:

  1. Draw two arrows.

  2. Label one as deforming force (FD) and the other as restoring force (FR).

  3. State whether the object was stretched or compressed.

Bodies: Perfect Elastic vs. Perfect Plastic

Perfect Elastic

• Definition: Returns to its original shape after a deforming force has acted on it.

• Examples:

  • Rubber band

  • Golf ball

Perfect Plastic

• Definition: Does not return to its original shape after a deforming force has acted on it.

• Examples:

  • Clay

Examples

• Examples of Elastic vs. Plastic bodies:

  • Bow (from a bow and arrow): Perfect Elastic body

  • Spring: Perfect Elastic body

  • Chewing gum: Perfect Plastic body

  • Trampoline: Perfect Elastic body

  • Springboard: Perfect Elastic body

  • Bread Dough: Perfect Plastic body

  • Aluminum: Perfect Plastic body

Perfect Elastic vs. Perfect Plastic Summary

Elastic Limit

• Definition: The maximum force that can be applied to a body so that the body regains its original form completely upon removal of the force (page 122).

• Considerations:

  • How much force can be applied before the object breaks?

  • Force ≤ Elastic limit: Object returns to equilibrium.

  • Force > Elastic Limit: Object loses its original form.

Hooke's Law

• States that within the limit of elasticity, stress is directly proportional to strain (page 132).

• Before breaking point is reached.

• σ ∝ ε

• If one increases, the other increases.

• If one decreases, the other decreases.

• Graphical Representation:

  • The graph is linear within the elastic limit.

  • Modulus of Elasticity = Gradient = \frac{Δy}{Δx}

• If Hooke’s Law is no longer followed:

  • Bendy and recoverable under deformation.

  • Stiff and will easily break with deformation.

Stress

• Definition: The internal restoring force per unit area of the body (page 124).

• Considerations:

  • How big is the restoring force (Deforming = restoring)?

  • How big is the Body (Area)?

Stress Calculations

• Formula 1: Calculating stress

  • σ = \frac{F}{A}

  • σ: Stress (Pa) / (N \cdot m^2)

  • F: Deforming force/ force on surface (N)

  • A: Area (m^2)
    *Formula 1 is on formula sheet

• Formula 2: Calculating area for a rectangle

  • A = l × w

  • l: Length (m)

  • w: Width (m)

  • A: Area (m^2)
    *Formula 2 is NOT on formula sheet

• Formula 3: Calculating area for a circle

  • A = πr^2

  • r: radius (m)

  • A: Area (m^2)

  • Remember: diameter = 2 x radius
    *Formula 3 is NOT on formula sheet

Strain

• Definition: The ratio of the change in dimension of an object to the original dimension of the object (page 128).

• How much bigger or smaller is the object?

• How much deformation took place?

Strain Calculations

• Formula 1: Calculating strain

  • ε = \frac{Δl}{L}

  • Δl : Change in length (m)

  • L: Original Length (m)

  • ε: Strain (no unit)
    *Formula 1 is on formula sheet

• Formula 2: Calculating area for a rectangle

  • A = l × w

  • l: Length (m)

  • w: Width (m)

  • A: Area (m^2)
    *Formula 2 is NOT on formula sheet

• Formula 3: Calculating area for a circle

  • A = πr^2

  • r: radius (m)

  • A: Area (m^2)

  • Remember: diameter = 2 x radius
    *Formula 3 is NOT on formula sheet

Hooke’s Law Calculations

• Formula 1: Calculating the modulus for elasticity

  • K = \frac{σ}{ε}

  • σ: Stress (Pa)

  • ε: Strain (no unit)

  • K: Modulus for elasticity (Pa)
    *Formula 1 is on formula sheet

Examples

*See Transcript for Examples

Exercises

• Exercise 4.1 p.

• Questions 4.1, 4.3, 4.3, 4.5