Sturdy Guide: Newton's Law of Universal Gravitation and Kepler's

Newton's Law of Universal Gravitation

• Sir Isaac Newton approached the problem of gravity by examining the motion of heavenly bodies, specifically the Moon and planets, and the force required to maintain the Moon's nearly circular orbit around the Earth.

• Newton concluded that because falling objects accelerate, they must have a force exerted on them. This force is known as the force of gravity ($F_G$).

• Direction of Force: Newton observed that the force of gravity on objects at the Earth's surface is always directed toward the center of the Earth. He deduced that the Earth itself exerts this force.

• Newton's Inspiration: According to legend, seeing an apple fall from a tree inspired Newton to consider that gravity acts not just at the tops of trees or mountains, but perhaps all the way to the Moon.

• Action at a Distance: This concept was controversial among contemporary thinkers who preferred contact forces (e.g., a hand pushing a cart). Newton proposed that gravity acts without contact, even across vast distances.

• Centripetal Acceleration of the Moon:

  • Calculated using the formula $a_R = \frac{v^2}{r}$.

  • The resulting acceleration is $0.00272\,m/s^2$.

  • This can be expressed in terms of surface gravity ($g = 9.80\,m/s^2$) as:     $a_R = \frac{0.00272\,m/s^2}{9.80\,m/s^2} \approx \frac{1}{3600} g$

• Distance Relationship:

  • The Moon is approximately $384,000\,km$ from Earth.

  • The Earth's radius ($r_E$) is $6380\,km$.

  • Therefore, the Moon is $60$ times farther from the Earth's center than objects on the surface ($384,000 / 6380 \approx 60$).

  • Since $60^2 = 3600$, Newton concluded that the gravitational force decreases with the square of the distance ($r$) from the Earth's center:     $F_G \propto \frac{1}{r^2}$

• Mass Relationship: The force of gravity is directly proportional to the mass of the object and the mass of the Earth.

• Symmetry and Newton's Third Law: When the Earth exerts a force on the Moon ($F_{ME}$), the Moon exerts an equal and opposite force on the Earth ($F_{EM}$):   $F_{ME} = -F_{EM}$

The Statement of the Law and the Gravitational Constant

• Newton's Law of Universal Gravitation: Every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. The force acts along the line joining the two particles.

• Mathematical Formula:   $F_G = G \frac{m_1 m_2}{r^2}$

  • where $m_1$ and $m_2$ are the masses of the two particles.

  • $r$ is the distance between them.

  • $G$ is the universal gravitational constant.

• The Value of G:

  • Because the gravitational attraction between ordinary objects is extremely small, $G$ was first measured by Henry Cavendish in 1798, over 100 years after Newton published the law.

  • Cavendish used an apparatus to measure the small force between ordinary-sized objects, confirming Newton's hypothesis.

  • The accepted value today is $G = 6.67 \times 10^{-11}\,N \cdot m^2/kg^2$.

Vector Form and Superposition

• Vector Representation:   $\mathbf{F}_{12} = -G \frac{m_1 m_2}{r_{21}^2} \mathbf{\hat{r}}_{21}$

  • $\mathbf{F}_{12}$ is the vector force on particle 1 exerted by particle 2.

  • $\mathbf{\hat{r}}_{21}$ is the unit vector pointing from particle 2 toward particle 1.

  • The negative sign indicates that the force on particle 1 is directed toward particle 2 (attractive force).

• Displacement Vectors: The displacement vector $\mathbf{r}_{12}$ is equal in magnitude to $\mathbf{r}_{21}$ but points in the opposite direction: $\mathbf{r}_{12} = -\mathbf{r}_{21}$.

• Principle of Superposition: For a system of many particles, the total gravitational force on a given particle is the vector sum of the forces exerted by all other particles:   $\mathbf{F}_1 = \mathbf{F}_{12} + \mathbf{F}_{13} + \mathbf{F}_{14} + \dots + \mathbf{F}_{1n} = \sum_{i=2}^{n} \mathbf{F}_{1i}$

Gravity at the Earth's Surface and the Shell Theorem

• Weight as Gravitational Force: The weight of an object on Earth ($mg$) is the gravitational force exerted by Earth:   $mg = G \frac{m M_E}{r_E^2}$

• Solving for g: The acceleration of gravity at the surface is:   $g = G \frac{M_E}{r_E^2}$

• Distinction between G and g: $G$ is a universal constant, while $g$ is the acceleration due to gravity specific to a location/planet.

• Mass of the Earth: Using the measured values of $g = 9.80\,m/s^2$ and $r_E = 6.38 \times 10^6\,m$, Cavendish calculated the Earth's mass:   $M_E = \frac{g r_E^2}{G} = 5.98 \times 10^{24}\,kg$

• Newton's Shell Theorem:

  • A thin uniform spherical shell exerts a force on an external particle as if all the shell's mass were at its center.

  • A thin uniform shell exerts zero force on a particle located inside the shell.

• Internal Earth Gravity Example: If a particle is halfway to the Earth's center ($r = \frac{1}{2} r_E$), only the mass inside that radius exerts a net force. In a uniform Earth scenario, the mass inside is $1/8$ the total mass ($V \propto r^3$). The force is calculated as $(1/8) / (1/2)^2 = 1/2$ of the surface gravity.

• Non-Uniformity of Earth: The Earth is not a perfect sphere; it bulges at the equator and has varying mass distributions. Rotation also affects the value of $g$.

• Table of g Values:

  • New York (0 m): $9.803\,m/s^2$

  • San Francisco (0 m): $9.800\,m/s^2$

  • Denver (1650 m): $9.796\,m/s^2$

  • Pikes Peak (4300 m): $9.789\,m/s^2$

  • Equator (0 m): $9.780\,m/s^2$

  • North Pole (0 m): $9.832\,m/s^2$

• Gravity Anomalies: Small variations in $g$ (1 part in $10^6$ or $10^7$) are used by geophysicists to find mineral deposits (denser rocks increase $g$) or oil-bearing salt domes (lower density decreases $g$).

Satellites and Apparent Weightlessness

• Satellite Motion: A satellite is maintained in orbit by gravity, which acts as the centripetal force ($F_R$).

• Orbital Speed:   Applying Newton's second law ($\sum F_R = m a_R$):   $G \frac{m M_E}{r^2} = m \frac{v^2}{r}$   Solving for velocity:   $v = \sqrt{\frac{G M_E}{r}}$

• Orbital Characteristics:

  • The speed of a satellite depends only on the mass of the attracting center ($M_E$) and the distance ($r$), not the mass of the satellite ($m$).

  • All satellites at the same altitude orbit at the same speed and have the same orbital period.

  • Circular orbits require the least takeoff speed. High speeds are needed; if a satellite stopped, it would fall directly to Earth.

• Apparent Weightlessness:

  • In a satellite, objects and the spacecraft are in free fall together, accelerating at the same rate toward Earth.

  • This is compared to a freely falling elevator where a scale would read zero ($w' = 0$) because the floor drops at the same rate as the object ($a = -g$).

  • Apparent Weight in an Elevator ($w'$):

  • At rest: $w' = mg$

  • Accelerating upward ($+a$): $w' = m(g + a)$

  • Accelerating downward ($-a$): $w' = m(g - a)$

  • Free fall ($a = -g$): $w' = mg - mg = 0$

• Real Weightlessness: Only occurs far out in space, distant from all attracting celestial bodies.

Kepler's Laws of Planetary Motion

• Historical Perspective:

  • Geocentric View: Sun and planets revolve around Earth (Ptolemy).

  • Heliocentric View: Earth and planets orbit the Sun (Copernicus, advanced by Galileo and Kepler).

• Kepler's Laws: Developed around 1600 based on data from Tycho Brahe (without telescopes).

  • Kepler's First Law: The path of each planet is an ellipse with the Sun at one focus.

  • Kepler's Second Law: A line from the Sun to the planet sweeps out equal areas in equal periods of time; planets move faster when closest to the Sun.

  • Kepler's Third Law: The ratio of the squares of the periods ($T$) of any two planets is equal to the ratio of the cubes of their semimajor axes ($s$):     $\frac{T_1^2}{T_2^2} = \frac{s_1^3}{s_2^3}$

• Newton's Derivation of Kepler's Third Law (Circular Case):

  • Set gravitational force equal to centripetal force: $G \frac{m_1 M_S}{r^2} = m_1 \frac{v^2}{r}$.

  • Substitute $v = \frac{2 \pi r}{T}$.

  • Rearranging yields: $\frac{T^2}{r^3} = \frac{4 \pi^2}{G M_S}$.

  • This shows that $\frac{T^2}{r^3}$ is a constant for all objects orbiting the same central mass.

• Perturbations: Observed deviations from Kepler's Laws occur because planets attract each other. Deviations in Uranus's orbit predicted the existence and location of Neptune. Similarly, Pluto was discovered by studying Neptune's orbit.

• Extrasolar Planets: Detected via the "wobble" in stars' positions caused by gravitational attraction from orbiting planets.

Gravitational Field and Fundamental Forces

• The Field Concept: To resolve the "action at a distance" dilemma, Michael Faraday developed the field concept. Every mass creates a gravitational field in the space surrounding it.

• Definition of Gravitational Field (g): The gravitational force per unit mass at a point in space.   $\mathbf{g} = \frac{\mathbf{F}}{m}$

• Units: $N/kg$ (equivalent to $m/s^2$).

• Magnitude Near a Single Mass (M):   $g = \frac{G \frac{m M}{r^2}}{m} = G \frac{M}{r^2}$

• Fundamental Forces in Nature:

  1. Gravitational Force: Acts between masses.

  2. Electromagnetic Force: Related to electric and magnetic charges; responsible for contact forces (pushes, friction).

  3. Strong Nuclear Force: Holds protons and neutrons in the nucleus.

  4. Weak Nuclear Force: Involved in radioactivity.

• Unification Theories: Electroweak theory unites electromagnetic and weak forces. Grand Unified Theories (GUT) attempt to unite these with the strong force.

The Principle of Equivalence and General Relativity

• Inertial vs. Gravitational Mass:

  • Inertial Mass: Measure of resistance to any force ($F = ma$).

  • Gravitational Mass: Determines the strength of the gravitational force.

• Principle of Equivalence: Proposed by Einstein; stating that the two masses are indistinguishable and no experiment can distinguish between an acceleration and a gravitational force.

• Bending of Light: According to the principle, gravity must act on light beams. If an elevator accelerates upward, a light beam entering a hole appears to curve downward. By equivalence, a gravitational field must also bend light.

• Experimental Confirmation:

  • Einstein's general theory of relativity predicted stars near the Sun would appear shifted by $1.75"$ of arc.

  • This was confirmed during a 1919 solar eclipse.

• Curvature of Space: Massive objects cause space (space-time) to curve. This is often visualized using a rubber-sheet metaphor where a weight causes a depression.

• Black Holes: Stars so dense and massive that the space-time curvature is so extreme that light cannot escape. Evidence suggests a massive black hole exists at the center of our Galaxy.

Moon Phases and Reference Frames

• Daily Moon Cycle: The Moon rises approximately $50\,min$ later each day.

• Synodic Period: Average time between successive full moons (relative to Earth), which is $29.53\,days$.

• Sidereal Period: The time for one full revolution around Earth relative to the stars, which is $27.32\,days$.

• Cause of Difference: As the Moon orbits Earth, Earth moves in its own orbit around the Sun. The Moon must travel an extra 2 days to realign with the Sun and Earth for a full moon phase.

• Relativity of Motion: While we describe the Earth as orbiting the massive Sun for simplicity (heliocentric), one can describe motion from Earth's frame (geocentric) where the Sun orbits the Earth with a period of 24 hours.