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
Define the system as the object.
Draw a free body diagram showing:
Tension force (upward).
Gravitational force by Earth (downward).
Apply momentum principle.
Set net force to zero for calculations:
Tension (upward) = gravitational force (downward).
Calculation: T = mg = 0.5 kg * 9.8 N/kg = 4.9 N (bottom tension).
Steps to Calculate Top Tension
Define the system as both wire and object.
Draw free body diagram including:
Force by support (upward).
Gravitational force by Earth (downward).
Apply momentum principle:
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
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.
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.