Newton's Law of Gravitation and Gravitational Fields: Review and Applications

Newton's Law of Gravitation

Fundamental Concepts

Newton's Law of Universal Gravitation describes the attractive force between any two masses. This law is foundational for understanding planetary motion, gravitational fields, and the behavior of objects under gravity.

• Newton's Law of Gravitation: The gravitational force FG between two point masses m1 and m2 separated by a distance r is given by:

$F_G = G \frac{m_1 m_2}{r^2}$

• Gravitational Constant (G): $G = 6.674 \times 10^{-11} \ \mathrm{N\,m^2/kg^2}$

• Direction: The force acts along the line joining the centers of the two masses.

• Application: Used to calculate the force between planets, stars, and other celestial bodies.

Gravitational Force Outside and Inside a Sphere

Outside a Uniform Sphere

For an object of mass m located outside a uniform sphere of mass M and radius R, the net gravitational force is:

$F_G = G \frac{M m}{R^2}$

• All mass acts as if concentrated at the center.

• Example: Calculating the force on a satellite orbiting Earth.

Inside a Uniform Sphere

When the object is inside the sphere, only the mass enclosed within radius r contributes to the gravitational force.

• Partial Mass: $m_p = \frac{M}{R^3} r^3$

• Gravitational Force: $F_G(r) = G \frac{m_p m}{r^2} = G \frac{M m}{R^3} r$

• Force is proportional to distance from the center: $F_G \propto r$

• Example: Force on an object at half the radius of Earth.

Partial Mass Gravitational Force

Derivation of Partial Mass

For a planet of uniform density, only the mass within radius r affects the gravitational force on an object at that location.

• Volume of sphere of radius r: $V = \frac{4}{3} \pi r^3$

• Density: $\rho = \frac{M}{\frac{4}{3} \pi R^3}$

• Partial mass: $m_p = \rho \frac{4}{3} \pi r^3 = \frac{M}{R^3} r^3$

Gravitational Force as a Function of Position

Graphical Representation

The gravitational force on an object as it moves from the center of a sphere to outside the sphere changes as follows:

• Inside the sphere ($0 < r < R$): $F_G(r) = G \frac{M m}{R^3} r$ (linear increase)

• Outside the sphere ($r > R$): $F_G(r) = G \frac{M m}{r^2}$ (inverse square law)

• Graph: Force increases linearly inside, peaks at the surface, then decreases as $1/r^2$ outside.

Gravitational Force Inside a Shell

Newton's Shell Theorem

For an object inside a spherical shell of uniform mass, the net gravitational force is zero.

• Shell Theorem: The gravitational forces from all parts of the shell cancel out inside.

• Implication: No net force acts on an object inside a hollow shell.

• Example: An astronaut inside a hollow planet would feel no gravitational pull from the shell.

Gravitational Field Graphs and Applications

Gravitational Field of Spherical Objects

The gravitational field g at a distance r from the center of a spherically symmetric object is:

• Thin Spherical Shell: $g = G \frac{M}{r^2}$ for $r > R_0$; $g = 0$ for $r < R_0$

• Solid Sphere: $g = G \frac{M}{R^3} r$ for $r < R$; $g = G \frac{M}{r^2}$ for $r > R$

Gravitational Field of a Planet

For a planet with mass $2M$ and radius $\frac{1}{2}R$:

$g_p = \frac{G (2M)}{(\frac{1}{2}R)^2} = \frac{2G M}{\frac{1}{4} R^2} = 8 \frac{G M}{R^2} = 8g$

• Field strength increases with mass and decreases with the square of the radius.

Gravitational Field of Sphere and Shell

For a sphere of radius R and mass M concentric with a shell of radius 3R and mass M, at a distance 2R from the center:

$g = \frac{G M}{(2R)^2} = \frac{G M}{4R^2}$

• Only the mass inside the radius contributes to the field at that point.

Gravitational Field of a Rocket

For a rocket of mass m launched from a planet of radius R and mass M to a distance 2R above the surface:

• Field strength at orbit: $g = \frac{G M}{(3R)^2} = \frac{G M}{9R^2}$

• Object's mass does not affect the field strength; only the planet's mass and distance matter.

Summary Table: Gravitational Force and Field Equations

Situation	Force Equation	Field Equation
Outside Sphere	$F_G = G \frac{M m}{r^2}$	$g = G \frac{M}{r^2}$
Inside Sphere	$F_G = G \frac{M m}{R^3} r$	$g = G \frac{M}{R^3} r$
Inside Shell	$F_G = 0$	$g = 0$
Thin Shell (outside)	$F_G = G \frac{M m}{r^2}$	$g = G \frac{M}{r^2}$

Key Takeaways

• Newton's Law of Gravitation applies universally to all masses.

• Inside a sphere, only the enclosed mass contributes to the force.

• Inside a shell, the net gravitational force is zero (Shell Theorem).

• Gravitational field strength depends on mass and distance.

• Applications: Planetary motion, satellite orbits, gravitational field calculations.

Additional info: These notes expand on the original slides by providing full derivations, definitions, and context for the equations and concepts presented.