The Forces

The four fundamental forces are:

1. Gravitational Force: Acts between masses, infinite range, weakest of the four forces but dominant at cosmic scales
2. Electromagnetic Force: Acts between charged particles, infinite range, responsible for atomic structure and most everyday contact forces
3. Nuclear Force: Acts between protons and neutrons, extremely short range (~10^-15 m), strongest at small distances, holds nuclei together
4. Weak Force: Operates inside subatomic particles, extremely short range (smaller than nuclear force), responsible for beta decay and certain particle transformations

Note: While gravitational and electromagnetic forces dominate macroscopic interactions, nuclear and weak forces are crucial at subatomic scales.

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.

Conditions for validity of classical physics:

1. Size condition: Objects must have linear dimensions > 10^-6 m

   • At smaller scales: Quantum physics applies, particle-wave duality emerges

2. Velocity condition: Objects must move at speeds < 10^8 m/s

   • At higher speeds: Relativistic mechanics applies, time dilation and length contraction occur

At the boundaries:

• Classical approximations become increasingly inaccurate
• Quantum effects become observable (uncertainty, tunneling, quantization)
• Relativistic effects become significant (mass increases, time slows)
• Nuclear and weak forces become relevant at subatomic scales
• The very concept of "particle" breaks down at quantum scales

Classical physics is most accurate for everyday macroscopic objects moving at ordinary speeds, where gravitational and electromagnetic forces dominate.

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.

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.

• Vector addition principles:

  • Each force must be treated as a vector with magnitude and direction
  • Forces between same charges (p-p, e-e) point away from each other
  • Forces between opposite charges (p-e) point toward each other
  • All 12 force vectors must be summed vectorially

• Key considerations:

  • Distance dependency: Forces follow inverse square law (∝ 1/r²)
  • Relative positions: As atoms move, force vectors change in magnitude and direction
  • Cancellation effects: Some components of forces cancel due to symmetry

• Electromagnetic interaction aspects:

  • These forces represent the electromagnetic force, one of the four fundamental forces
  • At atomic scales, electromagnetic forces dominate over gravitational forces
  • The balance between attractive and repulsive components determines stability

• Application example: In H₂ molecule formation, the net attractive force overcomes repulsion at certain distances, allowing chemical bonding

Four types of forces can act between hydrogen atom particles:

1. Electromagnetic forces:

   • Between proton-proton (repulsive)
   • Between electron-electron (repulsive)
   • Between proton-electron (attractive)

2. Gravitational forces: Act between all particles with mass, but negligible at atomic scale

3. Nuclear forces: Only significant at extremely short ranges (~10^-15 m) within nuclei

4. Weak forces: Only relevant in nuclear decay processes

In typical hydrogen atom interactions, electromagnetic forces dominate, with 12 total force vectors existing between the four particles (each particle exerts force on the other three).

Note: The electrostatic interaction between the electron of one atom and the proton of another atom can lead to van der Waals forces, which are responsible for molecular bonding.

In a system of two hydrogen atoms:

• Total electrostatic force vectors: 12 vectors
• Calculation: 4 particles × 3 interactions per particle = 12 force vectors

This is significant because:

1. Each particle (2 protons + 2 electrons) interacts with all other three particles
2. These forces must be vector-summed to determine net forces on each particle
3. The balance of these forces determines:
   • Stability of molecular hydrogen formation
   • Potential energy of the system
   • Equilibrium separation distance

Understanding these 12 force vectors is crucial for explaining phenomena like:

• Molecular bond formation
• Hydrogen molecule stability
• Electron cloud distributions in molecular orbitals

This complex force interplay demonstrates why atomic interactions cannot be simplified to single-particle models when atoms approach each other.

General Expression

For two parallel charged rods of length l with +q and -q at opposite ends, separated by distance a, the electric force is:

$F = \frac{q^2}{4\pi\varepsilon_0}\left[\frac{1}{a^2} - \frac{2}{(l+a)^2} + \frac{1}{(2l+a)^2}\right]$

Key Limiting Cases

1. When rod length dominates (l >> a):

   • System approximates two point charges
   • Force simplifies to: $F \approx \frac{q^2}{4\pi\varepsilon_0 a^2}$
   • Force is attractive (toward the other rod)
   • Follows standard inverse-square Coulomb behavior

2. When separation dominates (a >> l):

   • Force approaches zero: F ≈ 0
   • Each rod appears as a dipole from far away
   • Attractive and repulsive components cancel out
   • Demonstrates the principle that electric dipoles have weak long-range interactions

Physical Understanding

• The four charge interactions (+q to +q, +q to -q, -q to +q, -q to -q) compete to determine net force
• The relative magnitudes of l and a determine which interactions dominate
• This illustrates how geometry affects electrostatic interactions between charge distributions