The Forces

Newton's Third Law of Motion: If a body A exerts a force $\vec{F}$ on another body B, then B exerts a force $-\vec{F}$ on A, with both forces acting along the line joining the bodies.

This law implies forces always exist in pairs because:

• Forces result from interactions between objects
• Every action force creates a simultaneous reaction force
• The action-reaction forces are equal in magnitude but opposite in direction
• Both forces are of the same type (e.g., both gravitational or both electromagnetic)
• The forces act on different objects (never on the same object)

Example: When a book rests on a table, the table exerts an upward force on the book, while the book exerts an equal downward force on the table.

Gravitational Force Equation for Spherically Symmetric Bodies: $F = G\frac{m_1m_2}{r^2}$

Where:

• $G = 6.67 \times 10^{-11}$ N⋅m²/kg²
• $m_1, m_2$ are the masses of the bodies
• $r$ is the distance between their centers

This special case is critically important because:

1. It allows us to treat spherically symmetric bodies as point masses located at their centers
2. It dramatically simplifies calculations involving planets, stars, and other astronomical bodies
3. It enables accurate modeling of Earth's gravitational field for practical applications
4. It provides the foundation for orbital mechanics and satellite motion
5. Without this simplification, gravitational calculations would require complex vector integration

This result can be proven using calculus by integrating the gravitational effects across all mass elements of both bodies.

Reason for dominance of gravity between neutral bodies:

1. Electromagnetic forces cancel out because:

   • Neutral bodies contain equal numbers of positive and negative charges
   • Attractive forces (between unlike charges) and repulsive forces (between like charges) precisely balance
   • The vector sum of all electromagnetic interactions approaches zero

2. Gravitational forces accumulate because:

   • Gravity is always attractive between masses
   • All mass elements contribute additively to the total gravitational force
   • No cancellation occurs since there is no "negative mass"

Example: Earth and a 1 kg object experience ~9.8 N of gravitational attraction, while their net electromagnetic force is effectively zero despite containing ~10^27 charged particles.

This explains why gravity, despite being inherently much weaker than electromagnetism (by factor ~10^36), dominates large-scale cosmic structure.

Coulomb force vs. Gravitational force:

Coulomb: $F_e = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}$ where $\frac{1}{4\pi\varepsilon_0} = 9.0 \times 10^9$ N⋅m²/C²

Gravitational: $F_g = G\frac{m_1m_2}{r^2}$ where $G = 6.67 \times 10^{-11}$ N⋅m²/kg²

If electrons had 1% less charge than protons:

• Matter would have net positive charge
• Neutral objects would repel with enormous electromagnetic forces
• For two 1 kg iron blocks at 1 m distance:
  • Net charge per block: ~4.3 × 10^5 C
  • Resulting Coulomb force: ~1.7 × 10^21 N
  • This force is ~10^20 times stronger than gravity
• Macroscopic objects could not form stable structures
• Planets, stars, and galaxies could not exist in their current form
• The universe would be dominated by electromagnetic repulsion rather than gravitational attraction

Forces between surfaces in contact:

1. Normal force:

   • Perpendicular to the contact surface
   • Repulsive (pushes surfaces apart)
   • Balances external forces attempting to push surfaces together

2. Frictional force:

   • Parallel to the contact surface
   • Opposes relative motion or tendency toward motion
   • Limited by coefficient of friction: $F_f ≤ μF_n$

Microscopic basis:

• At atomic level, surfaces never make perfect contact
• Actual contact occurs only at microscopic high points (asperities)
• At these contact points, atoms come close enough for their electron clouds to interact
• Repulsive electromagnetic forces between electron clouds create normal force
• Attractive electromagnetic forces between molecules create adhesion
• Frictional force arises from:
  1. Breaking and reforming electromagnetic bonds between surface atoms
  2. Deformation of asperities
  3. Interlocking of microscopic surface irregularities

Smoother surfaces have fewer points of contact but can develop stronger adhesion forces, explaining why extremely smooth surfaces can sometimes stick strongly together.

Key observations pointing to weak force as distinct:

1. Particle transformations:

   • Neutron (neutral) → proton (positive) + electron (negative) + antineutrino
   • Proton (positive) → neutron (neutral) + positron (positive) + neutrino
   • These transformations can't be explained by electromagnetic or nuclear forces alone

2. Unique characteristics:

   • Extremely short range (even shorter than nuclear force)
   • Much slower interaction rate than electromagnetic or nuclear forces
   • Violation of parity conservation (distinguishes between left and right)
   • Only force that can change a particle's "flavor" (type of quark)

3. Neutrino involvement:

   • Neutrinos interact almost exclusively through weak force
   • Extremely low interaction probability with matter
   • Beta decay energy spectrum is continuous rather than discrete (requiring neutrino)

The weak force operates within subatomic particles, enabling transformations that are fundamentally different from the interactions facilitated by other forces.

Relationship between g and G:

For an object near Earth's surface:

1. By Newton's Law of Gravitation, the force is: $F = G\frac{Mm}{r^2}$
2. By Newton's Second Law, the force is: $F = ma = mg$

Equating these expressions: $mg = G\frac{Mm}{r^2}$

Solving for g: $g = G\frac{M}{r^2}$

Where:

• G = 6.67 × 10^-11 N⋅m²/kg² (universal gravitational constant)
• M = 6 × 10^24 kg (Earth's mass)
• r = 6.4 × 10^6 m (Earth's radius)

Calculation: $g = 6.67 \times 10^{-11} \frac{\text{N}\cdot\text{m}^2}{\text{kg}^2} \times \frac{6 \times 10^{24} \text{ kg}}{(6.4 \times 10^6 \text{ m})^2} \approx 9.8 \text{ m/s}^2$

This shows that the acceleration due to gravity is determined by the mass and radius of the planet, and is independent of the mass of the falling object.

Contact forces between objects (like a book on a table) are manifestations of the electromagnetic force:

• When two surfaces touch, the charged constituents (electrons and protons) at the atomic level come extremely close
• The electromagnetic repulsion between electron clouds prevents objects from passing through each other
• These microscopic electromagnetic interactions manifest as macroscopic "contact forces"
• The normal force perpendicular to the surfaces is generally repulsive (pushing objects apart)
• The parallel component to the surfaces is known as friction
• Despite appearing as a different type of force, contact forces are fundamentally electromagnetic in origin

Example: A table supporting a book involves countless electromagnetic interactions between surface atoms creating a net upward force that exactly balances gravity.

• Total number: 12 electric force interactions
• Calculation: 4 particles (2 protons, 2 electrons) × 3 interactions per particle = 12 total interactions
• Each particle exerts forces on all particles in the other atom
• Significance: This demonstrates Newton's Third Law, as each force has a corresponding equal and opposite reaction force
• These electromagnetic forces follow Coulomb's Law: F = (1/4πε₀)(q₁q₂/r²)
• Example: The electron in the first atom experiences forces from both the electron and proton in the second atom

Note: This complex interaction network explains the stability of molecular structures and the basis for chemical bonding

As two hydrogen atoms transition from isolated to bonded:

1. Initial state (isolated atoms):

   • Electron-electron: Repulsive
   • Proton-proton: Repulsive
   • Electron-proton: Attractive (within each atom)
   • Net interaction: Minimal at large distances

2. Transition state (approaching atoms):

   • Electron clouds begin overlapping
   • Each electron experiences attraction to both protons
   • Electrons redistribute to maximize attractions while minimizing repulsions

3. Bonded state (molecular hydrogen):

   • Electrons occupy shared molecular orbital between both nuclei
   • Electron probability density increases in internuclear region
   • Nuclei experience effective shielding from each other

Principle explaining this change: Quantum mechanical exchange interaction and energy minimization. The system reaches lowest energy when electrons occupy a shared molecular orbital, allowing each electron to interact with both protons simultaneously.

This redistribution of electron density creates the covalent bond in H₂, with bond energy of 436 kJ/mol, demonstrating how electromagnetic forces transform through quantum effects to create stable molecules.