Gravitation and Planetary Motion: Study Notes for PHYS 120

Gravitation

Introduction to Gravitation

Gravitation is a fundamental force of nature responsible for the attraction between masses. It governs the motion of planets, satellites, and objects on Earth.

• Acceleration due to gravity (g): On Earth's surface, $g = 9.8\ \text{m/s}^2$.

• Universal Law of Gravitation: The gravitational force between two masses is given by Newton's law:

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

• G: Universal gravitational constant, $G = 6.67 \times 10^{-11}\ \text{N m}^2 \text{kg}^{-2}$.

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

Gravitational Force Examples

• Example: Calculating the gravitational force between a lion ($m_1 = 175\ \text{kg}$) and a gazelle ($m_2 = 40\ \text{kg}$) at $r = 3\ \text{m}$:

$F = \frac{6.67 \times 10^{-11} \times 175 \times 40}{3^2} = 5.2 \times 10^{-8}\ \text{N}$

• Comparison: The gravitational force is extremely small compared to the weight of the gazelle.

• Finding distance for force equal to weight:

Set $F_g = W$:

$\frac{G m_1 m_2}{r^2} = m_2 g \implies r = \sqrt{\frac{G m_1}{g}}$

For the lion and gazelle, $r \approx 3.45 \times 10^{-5}\ \text{m}$ (unrealistically close).

Satellite Motion

Orbital Motion and Gravity

Satellites orbit Earth due to the balance between gravitational force and centripetal force.

• Force balance: $\sum F = \frac{G m_e m}{r^2} = \frac{m v^2}{r}$

• Orbital speed: $v = \sqrt{\frac{G m_e}{r}}$

• Radius of orbit: $r$ is measured from Earth's center.

Geosynchronous Satellites

A geosynchronous satellite orbits Earth with a period equal to Earth's rotation (24 hours), remaining above the same point on the equator.

• Height above Earth's surface:

Set $T = 24\ \text{hours}$, $T = 86400\ \text{s}$.

$r^3 = \frac{G m_e}{4 \pi^2} T^2$

For Earth, $r \approx 4.2 \times 10^7\ \text{m}$, so height $h = r - r_e \approx 3.6 \times 10^7\ \text{m}$ (about 6 Earth radii above surface).

• Tangential speed: $v = \sqrt{\frac{G m_e}{r}}$

Apparent Weight

How Heavy Do You Feel?

Apparent weight is the normal force ($F_N$) you feel, which can differ from true weight ($mg$) if you are accelerating.

• At rest ($a = 0$): $F_N = mg$

• Accelerating upward ($a = \frac{1}{2}g$): $F_N = m(g + \frac{1}{2}g) = \frac{3}{2}mg$

• Accelerating downward ($a = -\frac{1}{2}g$): $F_N = m(g - \frac{1}{2}g) = \frac{1}{2}mg$

Laws of Planetary Motion

Kepler's Laws

Kepler's laws describe the motion of planets around the Sun:

1. First Law (Law of Ellipses): A planet's orbit is an ellipse with the Sun at one focus.

2. Second Law (Law of Equal Areas): A planet sweeps out equal areas in equal times.

3. Third Law (Law of Periods): The square of the orbital period is proportional to the cube of the semi-major axis:

$T^2 \propto a^3$

• Application: For Mars, whose orbit diameter is 1.52 times Earth's:

$\frac{T_M^2}{r_M^3} = \frac{T_E^2}{r_E^3}$

$T_M = T_E (1.52)^{3/2} = 1.87\ \text{years}$

Summary Table: Key Gravitational Equations

Concept	Equation	Description
Gravitational Force	$F_g = G \frac{m_1 m_2}{r^2}$	Force between two masses
Orbital Speed	$v = \sqrt{\frac{G m_e}{r}}$	Speed of satellite in circular orbit
Geosynchronous Orbit Radius	$r^3 = \frac{G m_e}{4 \pi^2} T^2$	Radius for 24-hour orbit
Kepler's Third Law	$T^2 \propto a^3$	Period and semi-major axis relationship
Apparent Weight	$F_N = m(g \pm a)$	Normal force felt under acceleration

Additional info:

• These notes cover essential concepts in gravitation, satellite motion, and planetary orbits, suitable for introductory college physics.

• Examples and equations are expanded for clarity and academic completeness.