Chapter 13: Gravitation – Structured Study Notes

Gravitation

Learning Goals

• Understand gravitational forces and their universal nature.

• Define and calculate the weight of an object.

• Analyze the speed, orbital period, and mechanical energy of satellites.

• Apply Kepler’s three laws to planetary motion.

• Describe the properties and detection of black holes.

Introduction to Gravitation

Fundamental Questions

• What governs the motion of particles in planetary rings?

• Why do celestial bodies like the moon and earth not fall into each other?

Newton’s Law of Universal Gravitation

Definition and Properties

• Law of Gravitation: Every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Mathematical expression: $F_g = G \frac{m_1 m_2}{r^2}$ where $F_g$ is the gravitational force, $G$ is the gravitational constant, $m_1$ and $m_2$ are masses, and $r$ is the distance between centers.

• Gravitational constant: $G = 6.67 \times 10^{-11} \ \mathrm{N \cdot m^2 / kg^2}$

• Gravitational forces are equal in magnitude and opposite in direction for any two interacting masses.

Gravitation and Spherically Symmetric Bodies

Mass Distributions

• The gravitational effect outside a spherically symmetric mass is as if all mass were concentrated at its center.

• For two spheres, the force between them is equivalent to the force between two point masses at their centers.

Gravitational Attraction in Astronomy

Galaxies and Solar Systems

• Galaxies contain billions of stars, gas, dust, and other matter.

• The mutual gravitational attraction binds all matter in a galaxy together.

Weight and Acceleration Due to Gravity

Definitions and Formulas

• Weight: The total gravitational force exerted on a body by all other bodies.

• At Earth's surface, weight is: $w = F_g = G \frac{m_E m}{R_E^2}$ where $m_E$ is Earth's mass, $m$ is the object's mass, and $R_E$ is Earth's radius.

• Acceleration due to gravity: $g = G \frac{m_E}{R_E^2}$

Walking and Running on the Moon

Effects of Reduced Gravity

• Transition from walking to running occurs when the ground's vertical force exceeds weight.

• On the moon, this transition happens at lower speeds due to lower gravity (objects weigh only 17% as much as on Earth).

• Apollo astronauts often ran even at slow speeds during moonwalks.

Weight Variation with Altitude

Altitude Effects

• Weight decreases as altitude increases above Earth's surface.

• Formula for weight at distance $r$ from Earth's center: $w = G \frac{m_E m}{r^2}$

• Earth's radius: $R_E = 6.37 \times 10^6$ m

Interior of the Earth

Density Distribution

• Earth is approximately spherically symmetric but not uniform in density.

• Density $\rho$ decreases with increasing distance $r$ from the center.

• Earth consists of a solid inner core, molten outer core, and mostly solid mantle.

Gravitational Potential Energy

Conservative Nature and Calculation

• Gravitational force is conservative; work done does not depend on the path taken.

• Change in gravitational potential energy: $\Delta U = -W_{grav} = U_1 - U_2$ General expression: $U = -G \frac{m_E m}{r}$

• If only Earth's gravity does work, total mechanical energy is conserved.

Dependence on Distance

• Gravitational potential energy becomes less negative as distance from Earth increases.

• For the Earth-astronaut system: $U = -G \frac{m_E m}{r}$

Escape Velocity and Satellite Motion

Escape Speed Calculation

• Minimum speed to escape Earth's gravity (neglecting air resistance, rotation, and moon's gravity): $v_{esc} = \sqrt{\frac{2GM_E}{R_E}}$

• Example: For $R_E = 6.37 \times 10^6$ m and $m_E = 5.97 \times 10^{24}$ kg, $v_{esc} \approx 1.12 \times 10^4$ m/s

Projectile and Satellite Trajectories

• The trajectory of a projectile depends on its initial speed.

• Satellites in orbit follow paths determined by their speed and altitude.

Circular Satellite Orbits

Orbital Mechanics

• For a circular orbit, satellite speed keeps its distance from Earth's center constant.

• Gravitational force provides the required centripetal acceleration: $F_g = m \frac{v^2}{r}$

• Satellites are in a state of apparent weightlessness because they are in free fall.

Kepler’s Laws of Planetary Motion

First Law: Elliptical Orbits

• Each planet moves in an elliptical orbit with the sun at one focus.

Second Law: Equal Areas

• A line from the sun to a planet sweeps out equal areas in equal times.

• Angular momentum of the planet about the sun remains constant (no torque).

Third Law: Period-Axis Relationship

• Orbital period $T$ is related to the semi-major axis $a$: $T=\frac{2\pi r^{3/2}}{\sqrt{Gm_S}}$ where $m_S$ is the mass of the sun.

• Period does not depend on eccentricity $e$.

Comet Halley

Structure and Behavior

• The nucleus is an icy body about 10 km across.

• Close to the sun, the nucleus evaporates, forming a long tail.

Planetary Motions and the Center of Mass

Two-Body Orbits

• Both the sun and planet orbit their common center of mass.

• The more massive body (sun) orbits closer to the center of mass.

Spherical Mass Distributions

Gravitational Interaction

• Gravitational force between two spherically symmetric masses is as if each mass were concentrated at its center.

• For a point mass outside a spherical shell, the shell acts as if all mass is at its center.

Point Mass Inside a Spherical Shell

• If a point mass is inside a spherically symmetric shell, the shell exerts no net gravitational force on it.

• Only the mass inside a sphere of radius $r$ exerts a net force.

Apparent Weight and Earth’s Rotation

Effects of Rotation

• Apparent weight varies with latitude due to Earth's rotation.

• At the poles, apparent weight equals true weight; at the equator, it is reduced by centrifugal effects.

Variation of g with Latitude and Elevation

Tabular Data

The value of $g$ (acceleration due to gravity) varies with latitude and elevation. See table below:

Station	North Latitude	Elevation (m)	g (m/s2)
Canal Zone	09°	0	9.78243
Jamaica	18°	0	9.78591
Bermuda	32°	0	9.79806
Denver, CO	40°	1638	9.79609
Pittsburgh, PA	40.5°	235	9.80118
Cambridge, MA	42°	0	9.80398
Greenland	70°	0	9.82534

Black Holes

Definition and Properties

• If a spherical nonrotating body has radius less than the Schwarzschild radius, nothing can escape from it.

• Such a body is called a black hole.

• Schwarzschild radius: $R_S = \frac{2GM}{c^2}$ where $M$ is the mass of the black hole, $c$ is the speed of light.

• The surface at $R_S$ is called the event horizon.

• Light cannot escape from inside the event horizon, so events inside are not observable.

Detecting Black Holes

• Black holes can be detected by observing x-rays emitted from their accretion disks.

• Matter from a companion star is pulled into the accretion disk, heated, and emits x-rays.

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