module 5 lecture 10 :Gravitation, Projectile Motion, and Satellite Orbits

Gravitation

Universal Law of Gravitation

The universal law of gravitation describes the attractive force between any two masses in the universe. This force is always attractive and acts along the line joining the centers of the two masses.

• Definition: The gravitational force between two objects of mass $m_1$ and $m_2$ separated by a distance $d$ is given by:

$F = G \frac{m_1 m_2}{d^2}$

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

• Properties:

  • Gravity is a long-range force; it never truly disappears.

  • All objects with mass attract each other.

  • Earth's round shape is due to gravitational attraction pulling all parts toward the center.

• Cavendish Experiment: This experiment was used to measure the value of $G$.

Effects of Changing Masses

• If both masses are doubled, the gravitational force quadruples.

• If only one mass is doubled, the force doubles.

• The force of Planet Y on Planet X is equal in magnitude to the force of X on Y (Newton's Third Law).

Gravity on Other Worlds

• Mass vs. Weight: Your mass remains constant, but your weight depends on the gravitational acceleration of the planet.

• Example: On the Moon, $g = 1.62 \text{m/s}^2$.

Free Fall

• In free fall, the only force acting is gravity.

• Acceleration is constant and equal to $g$.

• Velocity changes during free fall.

Ocean Tides

Ocean tides are caused by the gravitational pull of the Moon (and to a lesser extent, the Sun) on Earth's oceans.

• Spring Tides: Occur during new and full moons when the effects of the Sun and Moon combine, producing higher tides.

• Neap Tides: Occur when the Moon is halfway between new and full, causing the Sun and Moon's effects to partially cancel, resulting in less pronounced tides.

Newton vs. Einstein: Theories of Gravitation

• Newton's Law: Gravity is an action-at-a-distance force, meaning changes are instantly felt across space.

• Einstein's General Theory of Relativity: Gravity is the result of the curvature of space-time caused by mass. Massive objects distort space-time, and smaller objects move along these curves.

• Experimental Evidence: Deflection of light by the Sun, precession of Mercury's orbit, and other measurements confirm general relativity.

Projectile Motion

Basic Concepts

Projectile motion refers to the motion of an object thrown or projected into the air, subject only to gravity.

• Without Gravity: The object follows a straight-line path.

• With Gravity: The object follows a curved, parabolic path.

• Independence of Motion: Horizontal and vertical motions are independent.

• Vertical Acceleration: Always $g$ downward.

• Horizontal Velocity: Remains constant (if air resistance is neglected).

• Time of Flight: The time to rise and fall is the same for symmetric projectile motion.

Key Equations

• Horizontal Distance (Range):

$R = \frac{v_0^2 \sin 2\theta}{g}$

• Maximum Range: Achieved at a launch angle of $45^\circ$.

• Velocity at Any Point: Use the Pythagorean theorem to combine horizontal and vertical components.

Example Problem

• Two balls are released from the same height, one dropped vertically and one fired horizontally. Both hit the ground at the same time (neglecting air resistance).

Satellite Motion

Satellites as Projectiles

A satellite is a projectile that falls around Earth rather than into it, due to its sufficient tangential velocity.

• Definition: A satellite is an object in orbit around a planet or other body.

• Condition for Orbit: The satellite's falling distance matches Earth's curvature.

• Horizontal Velocity: Must be great enough to ensure continuous free fall around Earth.

• Energy Conservation: In circular orbits, kinetic and potential energy remain constant; in elliptical orbits, they vary but their sum remains constant.

Types of Orbits

• Circular Orbit: Speed and distance from the center remain constant.

• Elliptical Orbit: Speed and distance vary; kinetic energy is greatest at closest approach (perigee), potential energy is greatest at farthest point (apogee).

Kepler's Laws of Planetary Motion

Kepler's First Law

• All planets move in elliptical orbits with the Sun at one focus.

Kepler's Second Law

• The radius vector from the Sun to a planet sweeps out equal areas in equal time intervals.

Kepler's Third Law

• The square of the orbital period ($T$) of any planet is proportional to the cube of the semimajor axis ($a$) of its orbit:

$T^2 \propto a^3$

Orbit Terminology

• Perihelion: Closest point to the Sun.

• Aphelion: Farthest point from the Sun.

• Perigee: Closest point to Earth (for Earth orbits).

• Apogee: Farthest point from Earth (for Earth orbits).

Summary Table: Gravitational Force Dependence

Change in Mass	Resulting Gravitational Force
Both masses doubled	Force quadruples
One mass doubled	Force doubles
Distance doubled	Force reduces to one-quarter

Additional info:

• Projectile motion and satellite motion are foundational topics in classical mechanics and are essential for understanding orbital dynamics and space exploration.

• Kepler's laws are empirical laws that describe planetary motion and are explained by Newton's law of gravitation.