Unit 7 Study Notes: Fundamental Principles of Gravitation

Universal Gravitational Constant (G)

$6.67 \times 10^{-11} \, \text{N}\cdot\text{m}^2/\text{kg}^2$

Force Equation for Gravity ($F_g$)

$Fg = G \frac{m1 m_2}{r^2}$

Inverse-Square Law

If you double the distance, the gravitational force drops to one-quarter.

Shell Theorem - Outside a uniform spherical shell

The shell acts as if all its mass were concentrated at a single point at its center.

Shell Theorem - Inside a hollow uniform shell

The net gravitational force on a particle placed inside is zero.

Principle of Superposition

The net force is the vector sum of the individual forces from multiple masses.

Gravitational Field ($g$)

The gravitational force experienced by a test mass divided by that mass.

Local acceleration due to gravity on Earth's surface

Approximately $9.8 \, \text{m/s}^2$.

Inside the Sphere Gravitational Force Equation

$g_{inside} = \frac{GM}{R^3} \cdot r$.

Gravitational Potential Energy Formula ($U_g$)

$Ug = -\frac{Gm1m_2}{r}$.

Change in Potential Energy (calculus)

$\Delta U = - \int{ri}^{r_f} \vec{F} \cdot d\vec{r}$.

Negative Sign in Gravitational Potential Energy

Indicates that gravity is an attractive force.

Total Mechanical Energy in Orbits

$E{mech} = K + Ug = \frac{1}{2}mv^2 - \frac{GMm}{r}$.

Total Energy of a Satellite in Circular Orbit

$E_{total} = -\frac{GMm}{2r}$.

Mistake: Forcing $F = \frac{GMm}{r}$

Common error; it should be $F = \frac{GMm}{r^2}$.

Common Mistake with $r$ in Gravitational Equations

$r$ is the center-to-center distance; use $r = R + h$ for altitude above Earth's surface.

Difference Between Vectors and Scalars in Gravity

Force and Field are vectors; Energy and Potential are scalars.

Gravitational Force Direction

Gravity is always attractive; it only pulls, never pushes.

Newton's Third Law and Gravity

Forces between two masses are equal in magnitude and opposite in direction.

Mechanical Energy Conservation in Space

Total mechanical energy is conserved in a system without drag.

Gravitational Field Strength Outside a Sphere

Behaves like a point mass, $g = \frac{GM}{r^2}$.

Change in Gravitational Potential Energy

Negative energy indicates a bound system.

Mass Enclosed within Radius $r$ Inside a Sphere

$M_{enc} = M \frac{r^3}{R^3}$.

Potential Energy as a Scalar Quantity

It can be summed algebraically from multiple mass interactions.

Key Takeaway Inside a Uniform Solid Sphere

Gravity increases linearly with distance from the center.