Comprehensive Study Notes on Gravitation

Introduction to Gravitation

• Early Observations and History:

  • Humans become aware of the tendency of material objects to be attracted toward the Earth early in their lives. Examples include objects falling when thrown up, uphill movement being more tiring than downhill, and raindrops falling from clouds.
  • Galileo Galilei (1564–1642): The Italian physicist recognized that all bodies, regardless of their masses, are accelerated toward the Earth with a constant acceleration. He demonstrated this through experiments with bodies rolling down inclined planes and arrived at a value for acceleration due to gravity close to the modern accurate value.
  • Celestial Observations: Historically, humans observed that most stars have unchanging relative positions. However, planets showed regular motions against the background of stars.

• Evolution of Planetary Models:

  • Geocentric Model: Proposed by Ptolemy approximately 2,000 years ago. It suggested that all celestial objects (stars, Sun, planets) revolved around the Earth in circular orbits. It involved complicated schemes where planets moved in small circles whose centers moved in larger circles.
  • Indian Astronomers: Similar theories were advanced about 400 years after Ptolemy.
  • Aryabhatta (5th Century A.D.): Mentioned a heliocentric model in his treatise where the Sun is the center around which planets revolve.
  • Nicolaus Copernicus (1473–1543): A Polish monk who proposed a definitive heliocentric model where planets moved in circles around a fixed central Sun. This was initially discredited by the Church.
  • Tycho Brahe (1546–1601): A Danish nobleman who spent his lifetime recording naked-eye observations of the planets.
  • Johannes Kepler (1571–1640): Brahe's assistant who analyzed the data and extracted three elegant laws of planetary motion.

Kepler’s Laws

1. Law of Orbits: All planets move in elliptical orbits with the Sun situated at one of the foci of the ellipse.

   • This deviated from the Copernican model of circular orbits.
   • Ellipse Properties: A closed curve drawn such that for any point on it, the sum of distances from two fixed points (foci $F_1$ and $F_2$) is constant.
   • Perihelion ($P$): The point in the orbit closest to the Sun.
   • Aphelion ($A$): The point in the orbit farthest from the Sun.
   • Semi-major axis ($a$): Half the distance of the major axis ($AP$). In a circle, foci merge at the center and the semi-major axis becomes the radius.

2. Law of Areas: The line joining any planet to the Sun sweeps equal areas in equal intervals of time.

   • This implies that planets move slower when they are farther from the Sun and faster when nearer.
   • Conservation of Angular Momentum: The law of areas is a consequence of angular momentum conservation, which is valid for any central force. A central force is directed along the vector joining the Sun and the planet.
   • The area swept ($\Delta A$) in time ($\Delta t$) is given by:     $\Delta A = \frac{1}{2} (r \times v \Delta t)$$\frac{\Delta A}{\Delta t} = \frac{1}{2} \frac{(r \times p)}{m} = \frac{L}{2m}$     where $L = (r \times p)$ is the angular momentum. Since $L$ is constant for a central force, the areal velocity ($\Delta A / \Delta t$) is constant.

3. Law of Periods: The square of the time period of revolution ($T$) of a planet is proportional to the cube of the semi-major axis ($a$) of the ellipse traced by the planet.

   • Mathematical Relation: $T^2 \propto a^3$
   • Confirmation Data (Table 7.1): Data for Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune shows that the quotient $Q = T^2 / a^3$ (in units of $10^{-34}\,y^2\,m^{-3}$) is approximately constant, ranging from $2.95$ to $3.01$.

Universal Law of Gravitation

• Newton's Insight: Legend says Newton was inspired by an apple falling from a tree. He reasoned that the same force causing the apple to fall provides the centripetal acceleration for the Moon.
• Centripetal Acceleration of the Moon ($a_m$):$a_m = \frac{V^2}{R_m} = \frac{4\pi^2 R_m}{T^2}$     where $R_m \approx 3.84 \times 10^8\,m$ and $T \approx 27.3\,\text{days}$. Substituting these values gives a value for $a_m$ much smaller than $g$ ($9.8\,m/s^2$).
• Inverse Square Law: Newton concluded that gravitational force decreases with the inverse square of the distance from the Earth's center:     $a_m ∑ \frac{1}{R_m^2}$; $g ∑ \frac{1}{R_E^2}$ yielding $\frac{g}{a_m} ∣ 3600$
• Formal Statement: Every body in the universe attracts every other body with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
  • Magnitude: $F = G \frac{m_1 m_2}{r^2}$
  • Vector Form: $\mathbf{F} = G \frac{m_1 m_2}{r^2} \mathbf{\hat{r}} = G \frac{m_1 m_2}{r^3} \mathbf{r}$     where $G$ is the universal gravitational constant and $\mathbf{\hat{r}}$ is the unit vector from $m_1$ to $m_2$. The force is attractive (along $-\mathbf{r}$).
• Newton's Third Law: The force on $m_1$ due to $m_2$ ($F_{12}$) is equal and opposite to the force on $m_2$ due to $m_1$ ($F_{21}$): $F_{12} = -F_{21}$.
• Superposition Principle: For a collection of point masses, the total force on one mass is the vector sum of the individual gravitational forces exerted by all other masses.

The Gravitational Constant (G)

• Experimental Determination: First performed by Henry Cavendish in 1798 using a torsion balance.
• Apparatus: A bar with small lead spheres ($m$) at the ends is suspended by a fine wire. Large lead spheres ($M$) are placed nearby. The gravitational attraction creates a torque.
• Calculations:
  • Force between spheres: $F = G \frac{Mm}{d^2}$
  • Torque: $\tau = F \times L$ (where $L$ is the length of the bar).
  • Restoring torque of the wire: \tau_{restoring} = Α ̑
  • At equilibrium: G \frac{Mm L}{d^2} = Α ̑
• Universal Value: The currently accepted value is $G = 6.67 \times 10^{-11}\,N\,m^2/kg^2$.

Acceleration Due to Gravity of the Earth

• Extended Bodies and Shell Theorems:
  1. The force between a hollow spherical shell of uniform density and a point mass outside is as if the entire mass of the shell is concentrated at its center.
  2. The force of attraction by a hollow spherical shell on a point mass situated inside it is zero.
• Acceleration ($g$) on the Surface:$g = \frac{F}{m} = \frac{G M_E}{R_E^2}$     Knowledge of $g$, $G$, and $R_E$ allows for the estimation of the Earth's mass ($M_E$). This is why it is said "Cavendish weighed the earth."
• Acceleration Below and Above Surface:
  • Above the Surface (at height $h$):$g(h) = \frac{G M_E}{(R_E + h)^2}$         For small $h$ ($h \ll R_E$):         $g(h) ∣ g × (1 - \frac{2h}{R_E})$
  • Below the Surface (at depth $d$):         Inside the Earth, only the smaller sphere of radius $(R_E - d)$ contributes to the gravitational force.         $g(d) = \frac{G M_E (R_E - d)}{R_E^3} = g × (1 - \frac{d}{R_E})$
  • Note: Acceleration due to gravity is maximum on the surface and decreases moving both upward and downward.

Gravitational Potential Energy

• Concept: Conservative force where work done is independent of path. Change in potential energy equals the work done by the force.
• Near Earth's Surface: If force is assumed constant ($mg$), for height change from $h_1$ to $h_2$:     $W_{12} = mg(h_2 - h_1)$     Potential energy at height $h$: $W(h) = mgh + W_0$ (where $W_0$ is the P.E. at the surface).
• Arbitrary Distance ($r$): For a particle at distance $r$ from center ($r > R_E$):     $W(r) = - \frac{G M_E m}{r} + W_1$
• Reference Point: Conventionally, the potential energy at infinity is set to zero ($W_1 = 0$). Then:     $V(r) = -\frac{G M_E m}{r}$
• System of Particles: The total potential energy of an isolated system is the sum of energies for all possible pairs of constituent particles.
  • Example 7.3: Potential energy of 4 particles ($m$) on a square of side $l$:     $W = -5.41 \frac{Gm^2}{l}$     Potential at the center: $U = -4\sqrt{2} \frac{Gm}{l}$.

Escape Speed

• Definition: The minimum initial speed required for an object to reach infinity (not fall back to Earth).
• Energy Conservation: Total initial energy ($K.E. + P.E.$) must be greater than or equal to the energy at infinity.     $\frac{1}{2} m v_i^2 - \frac{G M_E m}{R_E + h} ≥ 0$
• Formula at Surface ($h=0$):$v_e = \sqrt{\frac{2 G M_E}{R_E}} = \sqrt{2 g R_E}$
• Numerical Values:
  • Earth: $v_e ∣ 11.2\,km/s$.
  • Moon: $v_e ∣ 2.3\,km/s$. This low escape speed explains why the Moon has no atmosphere (gas molecules escape easily).

Earth Satellites and Energy

• Satellite Motion: Circular or elliptical orbits. Moon is the only natural satellite ($T ∣ 27.3\,\text{days}$). Artificial satellites are used for telecommunications, geophysics, and meteorology.
• Orbital Speed ($V$): Centripetal force is provided by gravity:     $\frac{m V^2}{(R_E + h)} = \frac{G m M_E}{(R_E + h)^2} \implies V = \sqrt{\frac{G M_E}{R_E + h}}$
• Time Period ($T$):$T = \frac{2\pi(R_E + h)}{V} = \frac{2\pi(R_E + h)^{3/2}}{\sqrt{G M_E}}$     Squaring gives: $T^2 = k (R_E + h)^3$ where $k = \frac{4\pi^2}{G M_E}$.
• Satellite close to Earth ($h ∣ 0$):$T_0 = 2π \sqrt{\frac{R_E}{g}} ∣ 85\,\text{minutes}$.
• Energy of a Satellite:
  • Kinetic Energy ($K.E.$): $\frac{G M_E m}{2(R_E + h)}$
  • Potential Energy ($P.E.$): $-\frac{G M_E m}{(R_E + h)}$
  • Total Energy ($E$): $E = K.E. + P.E. = -\frac{G M_E m}{2(R_E + h)}$
  • Note: Total energy is negative for bound systems. If $E ≥ 0$, the object escapes.

Points to Ponder

• Conserved Quantities: In gravitational motion, angular momentum and total mechanical energy are conserved. Linear momentum is NOT conserved.
• Weightlessness: Experienced by astronauts because both the spaceship and the astronaut are in "free fall" toward Earth, not because gravity is absent.
• Central Force: Gravitational force is central for point masses and spherically symmetric bodies. For non-symmetric rigid bodies, the force may not act along the line joining centers of mass.
• Shielding: Electrical forces can be shielded (Faraday cage), but gravitational shielding is impossible.
• Potential Choice: The addition of a constant to potential energy does not alter the gravitational force; only the difference in potential energy matters.