Long Summary

21.1 Gravitational fields

Gravitational Field: The gravitational field around a mass is a representation of the gravitational force around that mass, where gravitational attraction exists between any two masses. The force exerted by one mass on another is reciprocal and equal in magnitude but opposite in direction.
Gravitational Field Strength (g): Defined as the force per unit mass acting on a small test mass placed in the field: $g = \frac{F}{m}$ where
- F = gravitational force on the test mass,
- m = mass of the test object.
Unit of Gravitational Field Strength: Newton per kilogram (N/kg) or equivalently m/s².

Weight: The force of gravity on an object is defined by the equation: $F = mg$ where:
- F = gravitational force,
- m = mass of the object,
- g = gravitational field strength at that point.
Acceleration in Free Fall: An object in free fall accelerates at the rate of gravitational field strength, which can also be defined as:
$a = \frac{F}{m} = g$ .
Weightlessness: An object described as weightless is actually unsupported, as it is only acted on by the force of gravity alone.

Radial Field:
- Definition: Field lines are directed towards the center, resembling spokes of a wheel. The gravitational force on a small mass near a much larger mass is always directed toward the center of that mass.
- Example: The gravitational force of the Earth diminishes with increasing distance from its center.
Uniform Field:
- Definition: The gravitational field strength has the same magnitude and direction throughout. In a uniform field, field lines are parallel and equally spaced.
- Example: Over small distances relative to Earth's radius, the change in gravitational field strength is negligible, thus can be approximated as uniform.

Gravitational Potential (V): Defined as the work done per unit mass to move a small object from infinity to a point in a gravitational field:
$V = \frac{W}{m}$
Gravitational Potential Energy Change:
$\Delta E<em>{gpe} = m(V</em>2 - V_1) = m\Delta V$ .
Position of Zero Gravitational Potential Energy: Defined at infinity, where gravitational forces are negligible.
Example Scenario: When a rocket moves away from a planet, its gravitational potential energy increases. If it has a gpe meter set to zero at a far distance, it reads negative at the planet's surface since the rocket must perform work against gravitational force to escape.

Definition: Surfaces of constant potential where no work is needed to move along them.
Relation to Gravity: Gravitational potential gradient is the change in potential per meter of height, represented as $g = \frac{\Delta V}{\Delta r}$ with $g$ acting in the opposite direction to the potential gradient.
Example: A 1 kg mass raised 1 m gains 9.8 J of gravitational potential energy immediately above the Earth's surface. Over small distances, the potential gradient near the Earth is constant and equals 9.8 J/kg/m.

Newton's Law of Gravitation: States that every pair of masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers: $F = \frac{Gm<em>1m</em>2}{r^2}$ where:
- $G$ = universal gravitational constant, $6.67 \times 10^{-11} \text{Nm}^2/ ext{kg}^2$ .
Kepler's Third Law: Relates the period of orbit to the radius of orbit, consistent for all planets:
$T^2 \propto r^3$

Example Calculation: Determine the gravitational force between two point masses of 10 kg separated by 0.1 m. The approximate value calculated demonstrates the negligible force unless one of the masses is considerably large.

Gravitational forces act always as attractive forces.
The equation leads to important conclusions about the motion of planetary bodies and allows for gravitational force deductions.

Describe graphs of gravitational field strength against distance (r) for points outside a planet's surface.
Compare this with the graph of gravitational potential (V) against distance.
Explain the significance of the gradient on the V-r graph.

Gravitational Field around a Spherical Body: Significant implication that the gravitational field strength for points outside a planet is equivalent to that of a point mass located at the center:
- $g = \frac{GM}{r^2}$
Graphs for Gravitational Field Strength (g vs. r): Exhibits an inverse-square relationship, sharply decreasing as distance increases.
Behavior of Gravitational Field Inside a Planet: Gravitational force towards the center reduces, ultimately leading to zero at the core of massive bodies. Only the mass within a radius 'r' contributes to the gravitational field.

Calculate the gravitational field strength of Earth at the surface and at 1000 km altitude using the above equations.
Show practical implications with estimates of varying height affecting gravitational force experienced.

Geostationary Satellites: Satellites that orbit directly above the equator with a period of 24 hours, allowing constant positioning relative to Earth's surface.
Centripetal Forces: Analyzing the relationship between gravitational pull and the centripetal forces necessary to maintain stable orbits.

Satellite Motion Dynamics: Equate gravitational forces to centripetal forces for determining satellite speeds and altitudes necessary for stable motion.
Example Calculation: Calculate the radius required for a geostationary orbit and corresponding altitude above Earth's surface.

Kinetic and Potential Energy of Satellites: Total energy calculations take into account the kinetic energy of orbit and gravitational potential energy, leading to overall energy considerations of orbital mechanisms.
Total Energy Formula:
The total energy of a satellite in circular orbit is given by:
$E = -\frac{GMm}{2r}$
Characters describing variation in energy, and conditions leading to stable orbits: viable calculations and implications for satellite dynamics.