Titus 2/7

Chapter Four: Ball Spring Model of a Solid

Introduction

• Focus on the ball spring model as a way to visualize solids, particularly wires.

• Atoms within a solid are imagined as arranged in a simple lattice structure. Not all solids share the same arrangements.

Ball Spring Model Explanation

• Simplistic model likened to filling a box with balls (like basketballs or volleyballs) stacked on top of one another.

• Springs represent the bonds between atoms in a solid, emphasizing the connection between the balls (atoms).

Free Body Diagrams and System Analysis

• Mass is hung from a spring to observe forces acting on the system.

• Define the system as the object hanging from the spring. Surroundings consist of all external forces acting on the object.

• Forces acting on the system are documented through free body diagrams (isolate the object and show forces exerted).

Momentum Principle

• Apply the momentum principle (Δp = f_net Δt) to understand changes in momentum.

• Analyze forces:

  • Force by the spring upward (on the object).

  • Force by the spring downward (on the support).

Understanding Tension

• The force exerted by the spring on both ends is equal in magnitude, termed as tension.

• Important phrases:

  • "The spring is under tension"

  • "The tension in the spring" indicates the force exerted by the spring on each end.

Application to Wires

• Analyze a wire system shown as a one-dimensional ball spring model:

  • Tension is same at both ends of a thin wire with negligible mass.

  • If a wire has significant mass, the stretch at the bottom will be greater than the top due to increased load from below.

Example Calculation of Tension

• Consider a dense wire holding a mass:

  • Mass of wire: 0.05 kg

  • Mass of object: 0.5 kg

  • Find tension at both top and bottom.

Steps to Calculate Bottom Tension

1. Define the system as the object.

2. Draw a free body diagram showing:

   • Tension force (upward).

   • Gravitational force by Earth (downward).

3. Apply momentum principle.

4. Set net force to zero for calculations:

   • Tension (upward) = gravitational force (downward).

5. Calculation: T = mg = 0.5 kg * 9.8 N/kg = 4.9 N (bottom tension).

Steps to Calculate Top Tension

1. Define the system as both wire and object.

2. Draw free body diagram including:

   • Force by support (upward).

   • Gravitational force by Earth (downward).

3. Apply momentum principle:

4. Calculate net force for wire and object together:

   • Total mass = 0.5 kg (object) + 0.05 kg (wire) = 0.55 kg.

   • Tension = (0.55 kg) * (9.8 N/kg) = 5.39 N (top tension).

Comparing Tension at the Top vs. the Bottom

• Tension at the top (5.39 N) is greater than at the bottom (4.9 N).

• Reasoning:

  • Top tension accounts for weight of wire and object.

  • Bottom tension only supports the object.

Physical Insights on Stretch in Wires

• The stretch in a wire offers insights into the stiffness of interatomic bonds.

• When a load is applied, the wire stretches, reflecting the elastic properties of the atomic bonds.

Experiment with Copper Wire

• Target: Determine interatomic bond stiffness for a copper wire under load.

• Parameters:

  • Length of wire: 2 meters.

  • Cross-sectional area: 1 mm² = 1 x 10⁻⁶ m².

  • Applied force: 98 N causing stretch of 1.51 mm.

  • Calculate diameter of copper atom based on copper density and atom mass.

Calculation of Atomic Parameters

1. Volume of Copper Atom:

   • Use density (8.94 g/cm³) to find volume per mole (64 g for copper).

   • Convert to volume per atom using Avogadro's number.

2. Diameter Calculation:

   • Take the cube root of volume to find approximate diameter of a copper atom (~2.29 x 10⁻⁹ m).

Total Atoms in Copper Wire

• Estimate number of atoms along the length:

  • Length (2 m) divided by diameter (~2.29 x 10⁻¹⁰ m).

  • Result: ~8.73 x 10¹⁰ atoms.

Cross-sectional Area Calculation

• Area of one atom calculated using diameter, and total atoms per plane calculated from the total area divided by area taken up by atoms:

  • Result: ~1.91 x 10¹³ atoms.

Summary of Key Concepts

• Tension varies through the wire depending on the load it supports.

• Mechanical properties of materials can be linked to atomic interactions using models like the ball spring model.