Titus 2/12

Tension in a Wire

• Load and Stretch

  • A wire is hung with a load, causing a small stretch (approx. 1.5 mm).

  • This stretch allows us to determine stiffness using a spring model.

• Modeling the Wire as a Spring

  • The wire is treated as a single spring to understand its properties and behavior under load.

  • The stiff nature of the wire suggests high stiffness under tension.

Parallel Springs Model

• Breaking Down the Wire

  • The wire can be viewed as composed of several parallel springs (chains).

  • Each chain functions as an effective spring.

  • Effective Stiffness of Parallel Springs:

    • Given by the formula: Effective stiffness = n * stiffness of an individual spring (where n = number of springs).

  • To find the stiffness of one spring, rearrange the formula: stiffness of one spring = effective stiffness / n.

• Calculating Stiffness

  • Earlier calculations yield an effective stiffness of 3.398 x 10^(-9) N/m for a wire consisting of many parallel chains, indicating a very small stiffness for each individual spring due to the large number of springs.

  • Adding springs in series will yield smaller stiffness compared to springs in parallel, validating that multiple bonds or chains lead to reduced overall stiffness per chain.

Single Chain Consideration

• Analyzing a Single Chain's Bonds

  • A chain contains interatomic bonds connected in series, each with its own stiffness (KSI).

  • The effective stiffness for springs in series can be calculated using: effective stiffness = k / N (where k = stiffness of an individual bond and N = number of bonds in series).

  • The number of bonds in series is approximately 8.73 x 10^(10).

• Final Calculation

  • Calculate the stiffness of an individual interatomic bond using the formula: stiffness of a single bond = (number of bonds * effective stiffness). In this instance, yielding a result of approximately 29.7 N/m (~30 N/m), which resembles the stiffness of everyday springs.

Material Dependence of Stiffness

• Different Materials

  • If different materials for the wire (e.g., steel, aluminum) were analyzed, the stiffness would differ based solely on the material properties rather than dimensions or geometry of the wire.

  • Young's modulus, which describes a material's elasticity, is influenced by material composition alone, regardless of wire's shape.

Mechanics of Stress and Strain

• Stress and Strain

  • Tension applied causes the wire to experience stress, defined as force per unit area (Stress = Tension/Area).

  • Strain relates to the change in length divided by original length (Strain = ΔL / L).

  • Young's Modulus:

    • Defined as modulus = stress/strain, which connects macroscopic tension measurements to atomic bond stiffness; it is a material property independent of geometry.

    • Validates that regardless of length or diameter of the wire, Young's modulus remains constant for any given material.

Concept of Normal and Friction Forces

• Normal Force

  • Normal force arises from compression of springs at atomic levels, acting perpendicular to surfaces in contact.

  • Example: a brick resting on a table experiences an upward normal force from the table.

• Static Friction

  • Friction acts parallel to contact surfaces. Static friction varies based on external force applied and has a maximum threshold determined by the coefficient of static friction.

  • Maximum static friction is calculated as: static friction max = coefficient of static friction * normal force.

  • The effective force can lead to sliding if it exceeds this threshold.

Application of Momentum Principle

• Solving Problems with Momentum Principle

  • Draw scenarios, define system & surroundings, and develop free-body diagrams to guide problem-solving.

  • Ensure to apply momentum principles methodically: analyze forces in each dimension, especially in y-direction where applying gravitational force is essential.

  • Regularly practicing this methodology aids in developing intuition for solving complex problems.