Gravitation

SYLLABUS

Newton’s law of gravitation, Gravitational field strength, Gravitational potential; Gravitational potential energy, Variation in value of ‘g’ due to altitude and depth, Centre of mass and center of gravity, Motion of a satellite: Orbital velocity and time period of the satellite, Escape velocity, Potential and kinetic energy of the satellite, Geostationary satellite, GPS

What is This Chapter About?

Why does an apple fall from a tree? Why does the Moon orbit the Earth rather than flying off into space? Why do all planets move around the Sun in elliptical paths? For thousands of years these questions had no unified answer. Then Isaac Newton made one of the greatest intellectual leaps in history — he realised that the same force pulling the apple downward is the same force keeping the Moon in orbit. That force is gravity, and it acts between every pair of masses in the universe.

Chapter 7 of NEB Grade 11 Physics is one of the richest and most rewarding chapters in the entire course. It carries 10 teaching hours — the highest of any Grade 11 chapter — and covers everything from Newton’s law of gravitation to satellites, escape velocity, and the Global Positioning System. The ideas here connect directly to space science, technology, and our understanding of the cosmos.

Key Concepts You Will Learn

Newton’s Law of Gravitation

Every pair of masses in the universe attracts each other with a gravitational force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them:

[latex]F = \frac{Gm_1m_2}{r^2}[/latex]

where G = 6.674 × 10⁻¹¹ N m² kg⁻² is the universal gravitational constant. This inverse square law means that doubling the distance reduces the force to one quarter. The gravitational force is always attractive and acts along the line joining the two masses.

Gravitational Field Strength

The gravitational field strength (g) at a point in space is the gravitational force per unit mass experienced by a small test mass placed at that point:

[latex]g = \frac{F}{m} = \frac{GM}{r^2}[/latex]

. Near the Earth’s surface, this gives the familiar

[latex]g \approx 9.8 \text{ m/s}^2[/latex]

. The gravitational field strength is a vector quantity pointing toward the centre of the attracting mass.

Gravitational Potential and Potential Energy

Gravitational potential (V) at a point is the work done per unit mass in bringing a small test mass from infinity to that point:

[latex]V = -\frac{GM}{r}[/latex]

. The negative sign indicates that the gravitational field is attractive — energy must be supplied to move a mass away from a gravitating body.

Gravitational potential energy is the work done against gravity in bringing a mass m from infinity to a distance r:

[latex]U = -\frac{GMm}{r}[/latex]

. This is the total energy stored in the gravitational interaction between the two masses.

Variation of g with Altitude and Depth

The value of g is not constant everywhere. It decreases with increasing altitude above the Earth’s surface:

[latex]g_h = g\left(1 – \frac{2h}{R}\right)[/latex]

for small h. It also decreases with increasing depth below the surface:

[latex]g_d = g\left(1 – \frac{d}{R}\right)[/latex]

. At the centre of the Earth, g = 0. These variations have important practical implications for precise measurements and engineering.

Centre of Mass and Centre of Gravity

The centre of mass of a system is the point where the entire mass can be considered concentrated for the purposes of analysing translational motion. The centre of gravity is the point where the entire weight of the body acts. For uniform gravitational fields these two points coincide, but they differ in non-uniform fields — an important distinction for extended objects in space.

Orbital Velocity and Time Period of Satellites

A satellite orbits the Earth because the gravitational force provides the centripetal force for its circular motion. Setting

[latex]\frac{GMm}{r^2} = \frac{mv^2}{r}[/latex]

gives the orbital velocity:

[latex]v_o = \sqrt{\frac{GM}{r}}[/latex]

. The corresponding time period is:

[latex]T = 2\pi\sqrt{\frac{r^3}{GM}}[/latex]

— this is essentially Kepler’s third law derived from first principles.

Escape Velocity

Escape velocity is the minimum speed needed for an object to escape a planet’s gravitational field completely, reaching infinity with zero kinetic energy remaining. Using energy conservation:

[latex]v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}[/latex]

. For Earth, this is approximately 11.2 km/s. Escape velocity does not depend on the mass or direction of the projectile — only on the planet’s mass and radius.

Energy of a Satellite

A satellite in orbit has both kinetic energy

[latex]KE = \frac{GMm}{2r}[/latex]

and gravitational potential energy

[latex]PE = -\frac{GMm}{r}[/latex]

. The total mechanical energy is

[latex]E = -\frac{GMm}{2r}[/latex]

— negative, indicating a bound system. As a satellite moves to a higher orbit (larger r), its total energy increases (becomes less negative), but its kinetic energy actually decreases. This counterintuitive result is important for understanding orbital mechanics.

Geostationary Satellites

A geostationary satellite orbits the Earth at the same angular velocity as the Earth rotates, so it appears stationary relative to the Earth’s surface. For this, its orbital period must equal exactly 24 hours, placing it at a specific altitude of approximately 36,000 km above the equator. Geostationary satellites are used for weather monitoring, television broadcasting, and communications.

Global Positioning System (GPS)

The GPS consists of a network of satellites whose positions and timing signals allow a receiver on Earth to determine its precise location by comparing signals from multiple satellites simultaneously. Understanding the basic working principle of GPS — triangulation using the known positions and signal travel times of at least four satellites — connects the physics of gravitation directly to modern technology used every day.

Why This Chapter Matters for NEB Exams

Gravitation is the highest-weightage chapter in Grade 11 Physics. NEB board examinations consistently test derivations of orbital velocity, escape velocity, and time period of satellites. Numerical problems on gravitational force, potential energy, orbital speed, and escape velocity appear in every paper. Variation of g with altitude and depth is a frequently asked short-answer question. The energy of a satellite — particularly explaining why total energy is negative — is a common conceptual question.

Important Topics to Focus On

• Statement and mathematical form of Newton’s law of gravitation
• Gravitational field strength and its relation to g
• Gravitational potential and potential energy — derivations and sign conventions
• Variation of g with altitude:

[latex]g_h = g\left(1 – \frac{2h}{R}\right)[/latex]

Variation of g with depth:

[latex]g_d = g\left(1 – \frac{d}{R}\right)[/latex]

Centre of mass versus centre of gravity Derivation of orbital velocity:

[latex]v_o = \sqrt{\frac{GM}{r}}[/latex]

Derivation of time period of satellite:

[latex]T = 2\pi\sqrt{\frac{r^3}{GM}}[/latex]

Derivation of escape velocity:

[latex]v_e = \sqrt{2gR}[/latex]

Kinetic, potential, and total energy of a satellite Conditions for geostationary satellites Basic working principle of GPS Numerical problems on all of the above

Connection to Other Chapters

Gravitation builds on Chapter 4 (Dynamics) — Newton’s law of gravitation is an application of his broader force framework — and Chapter 6 (Circular Motion) — orbital motion is circular motion with gravity providing centripetal force. Chapter 5 (Work, Energy and Power) is used to derive escape velocity through energy conservation. In Grade 12, gravitational potential energy connects to thermodynamics through the concept of potential wells, and orbital mechanics connects to the broader study of angular momentum in Chapter 1 (Rotational Dynamics).