Gravitation

• Formula: $G = \frac{kdr^2}{4MmlD}$
• Experimental parameters:
  • $k$ = torsional constant of suspension wire
  • $d$ = linear displacement of light spot on scale
  • $r$ = distance between centers of heavy ball and small ball
  • $M$ = mass of heavy ball
  • $m$ = mass of small ball
  • $l$ = length of rod connecting small balls
  • $D$ = distance between mirror and scale
• Working principle: Gravitational torque ($Fl$) equals opposing torsion torque ($k\theta$)
• Angular measurement: $\theta = \frac{d}{4D}$ (light beam deflection doubles angle)

• Newton's revolutionary assumption: The laws of nature are the same for earthly and celestial bodies

• Historical context:

  • Before Newton, western belief held that heavenly bodies followed different laws than earthly objects
  • The heavenly realm was considered perfect and unchanging
  • Tycho Brahe initially disbelieved his observation of a new star due to this belief

• Significance:

  • Unified terrestrial and celestial physics
  • Allowed applying earthly experimental results to celestial phenomena
  • Led to understanding that same force causing an apple to fall also keeps the moon in orbit
  • Established principle of universality in physics
  • Enabled mathematical analysis of planetary motion using terrestrial physics

Key sources of error in the Cavendish experiment include:

1. Air currents and temperature fluctuations:

   • Minimize by conducting the experiment in a sealed, temperature-controlled environment
   • Use radiation shields to prevent heat transfer between masses

2. Electrostatic and magnetic interference:

   • Ground all components to eliminate static charges
   • Use non-magnetic materials for apparatus construction
   • Shield the setup from external electromagnetic fields

3. Torsion wire imperfections:

   • Carefully select high-quality quartz fibers with consistent elasticity
   • Calibrate the torsional constant independently
   • Allow sufficient time for fiber relaxation

4. Measurement precision:

   • Use long optical lever arms to amplify angular displacements
   • Implement digital image analysis for precise angle measurement
   • Take multiple measurements and apply statistical analysis

5. Environmental vibrations:

   • Mount apparatus on vibration-isolation tables
   • Perform experiment in basement locations
   • Use seismic monitoring to identify and exclude data during disturbances

Modern versions achieve precision of approximately 0.01% by addressing these error sources systematically.

The torsion balance method works by:

1. Creating a measurable torque from extremely weak gravitational forces
2. Converting angular displacement to force measurement using the known properties of a suspension wire
3. Balancing competing torques: gravitational attraction vs. torsional resistance

Process:

• Two small masses at ends of a suspended rod are attracted to larger external masses
• Gravitational force creates torque (F·l) trying to rotate the rod
• Wire resists with countertorque proportional to twist angle (kθ)
• At equilibrium: gravitational torque = torsional torque
• Measuring deflection angle allows calculation of force

This approach was necessary because:

• Gravitational force between laboratory-sized masses is extremely small (~10⁻¹⁰ N)
• Direct force measurement devices lacked sufficient sensitivity
• Torsion amplifies tiny forces into observable rotations
• Allows isolation from other forces (especially electrostatic) that would overwhelm gravity

The method's success relies on the precise mathematical relationship between angle and torque in torsion systems, making otherwise imperceptible gravitational effects measurable.

Major sources of error in Cavendish G measurements:

1. Air currents and vibrations:

   • Minimize by housing apparatus in vacuum chamber
   • Use multiple layers of shielding and vibration isolation
   • Conduct experiment in basement/stable environments

2. Electrostatic interference:

   • Ground all components properly
   • Control humidity to reduce static buildup
   • Use conducting shields around apparatus

3. Temperature variations:

   • Create thermally stable environment
   • Allow system to reach thermal equilibrium before measurement
   • Use materials with low thermal expansion coefficients

4. Measurement precision issues:

   • Use optical lever technique with laser instead of light beam
   • Implement digital imaging for precise angle measurement
   • Apply multiple measurement cycles for statistical averaging

5. Torsion wire imperfections:

   • Carefully characterize torsion constant k
   • Account for anelasticity (time-dependent response)
   • Use high-quality, uniform quartz fibers

Modern refinements include:

• Time-of-swing methods rather than static deflection
• Feedback systems to maintain null position
• Computer analysis of oscillation patterns

With careful technique, precision can reach parts per thousand, though absolute accuracy remains challenging due to systematic effects.

The gravitational potential of a uniform ring at point P relates to distance parameters through:

$V = -\frac{GM}{\sqrt{a^2 + r^2}}$

As point P moves along the axis:

• When $r$ increases (P moves away from the ring), $\sqrt{a^2 + r^2}$ increases, making the potential magnitude decrease toward zero
• When $r$ approaches zero (P approaches the center O), the potential approaches $-\frac{GM}{a}$
• When $r$ becomes very large compared to $a$, the potential approximates $-\frac{GM}{r}$, resembling a point mass

The potential is always negative, with maximum magnitude when P is at the center of the ring.

Key insight: Unlike a point mass, the potential doesn't approach infinity as you approach the ring, making the force finite even at the center.

The gravitational potential inside a uniform spherical shell remains constant because:

1. Mathematical explanation:

   • For an interior point at distance r from center, the potential equals -GM/a
   • This expression contains no r dependence, only the shell radius a
   • This results from the integration of the contributions from all mass elements over the shell

2. Physical principle:

   • This is a consequence of Newton's shell theorem
   • The net gravitational force at any point inside a spherical shell is exactly zero
   • Since force is the gradient of potential, zero force means constant potential

3. Conceptual understanding:

   • Mass elements on opposite sides of the shell exert forces that precisely cancel
   • The solid angle subtended by closer portions is smaller but their gravitational effect is stronger
   • The solid angle subtended by farther portions is larger but their gravitational effect is weaker
   • These effects exactly compensate throughout the interior

This principle is used in electrostatics (Faraday cage effect) and explains why the Earth's gravitational field would be zero if you could reach its center.

For external points (r > a):

• E = GM/r² (directed toward the center)
• Behaves as if all mass is concentrated at the center

For internal points (r < a):

• E = (GM/a³)r (directed toward the center)
• Field strength is directly proportional to distance from center
• At the center (r = 0), the field is zero due to symmetry
• At the surface (r = a), both formulas yield E = GM/a²

Note: The transition between these formulas is continuous at r = a, ensuring there's no discontinuity in the gravitational field at the sphere's surface

• Physical principle: The behavior differs due to how spherical shells contribute to the total gravitational field

For external points:

• All spherical shells (of any radius) contribute to the gravitational field
• Each shell's contribution is equivalent to a point mass at the center
• The total field is the sum of all shells' contributions: E = GM/r²

For internal points:

• Only shells with radius less than r contribute to the gravitational field
• Shells with radius greater than r produce zero net field inside (their gravitational effects cancel)
• The mass contributing to the field is proportional to r³: m = M(r³/a³)
• This yields the linear relationship: E = (GM/a³)r

This principle explains why hollow spherical shells produce zero gravitational field inside them, a key result in gravitational theory

If the fifth force hypothesis were confirmed:

1. The simple equivalence principle would require modification, as objects would experience a combined force of standard gravity plus the fifth force component.

2. The observed "gravitational mass" would become distance-dependent:

   • At short distances: $m_{grav} = m(1 + \alpha) \approx 0.993m$
   • At large distances: $m_{grav} = m$

3. Free-fall acceleration would depend slightly on composition if different materials couple differently to the fifth force.

4. General relativity would need revision, as it's built on the exact equivalence of inertial and gravitational mass.

5. Experimental tests of the equivalence principle would show violations at specific distance scales (near λ ≈ 200m).

6. Precision tests using different materials at different distances could explicitly map the fifth force's strength and range.

This would represent a fundamental shift in our understanding of gravity and potentially open the door to new physics beyond the current standard model.