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.

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.

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

Spring Force: $F = k|x-x_0| = k|\Delta x|$

Where:

• $k$ is the spring constant (force per unit displacement)
• $x_0$ is the natural length of the spring
• $x$ is the current length
• $\Delta x$ is the displacement from natural length

Conditions for Hooke's Law validity:

• Small extensions or compressions only
• Material remains in elastic region (no permanent deformation)
• Temperature remains relatively constant
• No hysteresis effects

Relation to electromagnetic interactions:

• Spring force originates from electromagnetic forces between atoms in the material
• When a spring is stretched/compressed, interatomic distances change
• This disturbs the equilibrium between attractive and repulsive electromagnetic forces between atoms
• The resulting net force tries to restore original atomic spacing
• Macroscopically observed as the spring's restoring force

This is why spring forces, contact forces, and tension forces are fundamentally electromagnetic in nature.

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.

The total gravitational force between two extended bodies requires vector addition of all individual particle-particle interactions:

1. For each particle in body 1, calculate the force exerted on it by each particle in body 2
2. Each individual force follows the formula: F = G(m₁m₂)/r²
3. Sum all these force vectors vectorially

If two bodies contain particles labeled i (in body 1) and j (in body 2), the force components would be labeled F_ij, where:

• First index (i) indicates the particle in the first body
• Second index (j) indicates the particle in the second body

For example, with 3 particles in each body, you would add 9 force vectors: F₁₁, F₁₂, F₁₃, F₂₁, F₂₂, F₂₃, F₃₁, F₃₂, F₃₃

For continuous bodies, this summation becomes an integration process.

For spherically symmetric bodies, calculation simplifies dramatically:

• The gravitational force between two spherically symmetric bodies equals the force between two point masses of equivalent mass located at their centers
• Formula: F = G(m₁m₂)/r²
• r = distance between the centers of the two bodies
• This applies only when one body is completely outside the other

This mathematical simplification (known as the Shell Theorem) allows us to treat planets, stars, and other spherically symmetric objects as point masses for gravitational calculations, without having to perform complex integrations across their entire volumes.

Example: Earth-Moon gravitational calculations can be done by treating both as point masses at their centers.

• 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

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.