GRAVITATION

Notes on Gravitation

7.1 Introduction

Gravitation is a fundamental force that attracts two bodies towards each other. Our first experiences with gravitation happen when objects fall back to the earth, illustrated by Galileo's findings that all objects fall at the same rate regardless of mass.

7.2 Kepler’s Laws

Kepler formulated three laws regarding planetary motion:

1. Law of Orbits: Planets move in elliptical orbits with the Sun at one focus.
2. Law of Areas: The line joining a planet to the Sun sweeps out equal areas in equal times.
3. Law of Periods: The square of the orbital period is proportional to the cube of the semi-major axis of the orbit.

These laws paved the way for Newton’s law of gravitation.

7.3 Universal Law of Gravitation

Newton's Universal Law states:

Every body in the universe attracts every other body with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

This can be expressed mathematically as:

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

where G is the gravitational constant ( 6.67 x 10^-11 N m²/kg²).

7.4 Gravitational Constant

The gravitational constant was first accurately measured by Henry Cavendish using a torsion balance in 1798.

7.5 Acceleration due to Gravity of the Earth

Gravity can be expressed as:

$$ g = \frac{GM}{R^2} $$

Here, M is the mass of the Earth, and R is the radius of the Earth. The gravitational acceleration decreases with altitude and can be expressed for heights or depths:

• Above Earth's Surface: $$ g(h) = \frac{GM}{(R + h)^2} $$
• Below Earth's Surface: $$ g(d) = g(0) \left(1 - \frac{d}{R} \right) $$

7.6 Escape Speed

The escape speed from the Earth's surface is given by:

$$ v_e = \sqrt{\frac{2GM}{R}} $$

This speed for the Earth is approximately 11.2 km/s. If an object is thrown with this speed or greater, it can escape the Earth’s gravitational influence.

7.7 Gravitational Potential Energy

Gravitational potential energy (U) between two masses can be calculated:

$$ U = - G \frac{m_1 m_2}{r} $$

7.8 Energy of an Orbiting Satellite

The energy of a satellite in orbit comprises its kinetic and potential energy:

• Kinetic Energy (KE) is given as: $$ KE = \frac{1}{2} mv^2 $$
• Potential Energy (PE) due to gravity: $$ PE = - G\frac{m M}{R} $$

For circular orbits, total energy is negative indicating the system is bound.

7.9 Earth Satellites

Satellites orbiting the Earth follow Kepler’s laws. The period of orbit can be calculated using:

$$ T^2 = \frac{4\pi^2}{G M}R^3 $$

For artificial satellites, the time period and radius can effectively be utilized to understand their motion and properties.

7.10 Key Concepts of Gravitation

1. Gravitational force is an attractive force.
2. Kepler’s First Law states that planets move in ellipses.
3. Kepler’s Second Law involves equal areas swept out in equal time.
4. Acceleration due to gravity varies with height and depth.
5. Escape velocity depends only on the mass and radius of the celestial body.
6. Gravitational potential energy is negative and varies with the inverse of distance.
7. The concept of energy conservation is fundamental in understanding motion orbits in gravitational fields.
8. Satellite motion is governed by the same principles that govern planets around the sun.

1. Gravitational force is attractive and depends on mass and distance.
2. Kepler’s laws describe planetary motion in elliptical orbits.
3. Escape speed from Earth is about 11.2 km/s.
4. Gravitational potential energy is negative and becomes zero at infinity.
5. Acceleration due to gravity decreases with height and is affected by depth.
6. Newton’s universal law relates forces to mass and distance.
7. Circular orbits yield constant centripetal force from gravity.
8. Conservation of energy applies in gravitational systems.