Apply Gravitational Fields — Apply D.1 Gravitational fields...

SECTION 1

Historical Context & Motivation

Humans have wondered about falling objects and the motions of planets for thousands of years. Ancient Greek thinkers like Aristotle believed heavier objects fell faster, and that the Earth sat motionless at the centre of the universe. It took centuries of careful observation and bold thinking to overturn these ideas. The story of gravitational fields is really the story of how we learned that the same force pulling an apple to the ground also keeps the Moon in orbit.

1543

Copernicus and the Heliocentric Model

Nicolaus Copernicus published De Revolutionibus, arguing that the Earth and planets orbit the Sun, not the other way around. This shift set the stage for a mathematical description of celestial motion.

1609

Kepler's Laws of Planetary Motion

Johannes Kepler used Tycho Brahe's precise observations to show that planets follow elliptical orbits and sweep out equal areas in equal times. These empirical laws begged for a deeper explanation—what force produces such motion?

1687

Newton's Principia

Isaac Newton published the Principia Mathematica, presenting the law of universal gravitation: every mass attracts every other mass with a force proportional to their masses and inversely proportional to the square of the distance between them.

1798

Cavendish Measures G

Henry Cavendish used a delicate torsion balance to measure the gravitational constant G, effectively 'weighing the Earth' and confirming Newton's formula with a precise numerical value.

1915

Einstein's General Relativity

Albert Einstein re-imagined gravity not as a force but as the curvature of spacetime caused by mass and energy. For most IB-level problems, Newton's model is perfectly sufficient, but general relativity corrects it at extreme speeds and strong fields.

The central question this topic addresses is: How can we use a single equation—Newton's law of universal gravitation—to predict and explain everything from the weight of an astronaut to the period of a satellite? By the end of this lesson, you will be able to apply gravitational field concepts to solve a wide range of IB Physics problems.

SECTION 2

Core Principles & Definitions

Before diving into calculations, you need a solid grip on the key ideas behind gravitational fields. A gravitational field is the region of space around any mass in which another mass would experience a gravitational force. Fields are invisible, but their effects are very real—they dictate how objects accelerate, orbit, and fall.

Newton's Law of Universal Gravitation

Every point mass attracts every other point mass with a force F = Gm₁m₂ / r². The force is always attractive and acts along the line joining the two masses.

Gravitational Field Strength (g)

The gravitational field strength at a point is the force per unit mass experienced by a small test mass placed there: g = F / m = GM / r². Its SI unit is N kg⁻¹ (equivalently m s⁻²).

Gravitational Potential (V)

Gravitational potential at a point is the work done per unit mass to bring a test mass from infinity to that point: V = −GM / r. It is always negative, reflecting that gravity is attractive.

Field Lines & Equipotentials

Gravitational field lines point radially inward toward a mass. Equipotential surfaces are perpendicular to field lines. No work is done moving a mass along an equipotential surface.

Principle of Superposition

When multiple masses are present, the total gravitational field at any point is the vector sum of the individual fields. This lets you analyse systems with two or more bodies.

✦ KEY TAKEAWAY

Think of a gravitational field like a 'gravitational slope' surrounding every mass. Just as a ball rolls downhill without anyone pushing it, a mass 'rolls' toward a more massive object because the gravitational field creates a natural tendency toward the centre. The steeper the slope (stronger the field), the faster the acceleration. Moving away from the mass is like climbing uphill—you must do work against gravity, which is why gravitational potential becomes less negative farther out.

SECTION 3

Visualising Gravitational Fields

A good diagram can make abstract concepts concrete. The SVG below shows a central mass with radial gravitational field lines, concentric equipotential surfaces, and how gravitational field strength g decreases with distance according to the inverse-square law. Notice how the field lines point inward and the equipotential lines are circles (or spheres in 3D) perpendicular to them.

The cyan arrows represent gravitational field lines pointing radially toward the central mass M. The dashed violet circles are equipotential surfaces. Notice that V₁ (closest) is the most negative potential. As distance increases, the field lines spread apart, visually showing the inverse-square weakening of g.

A few important features to notice. First, the field lines always point toward the mass because gravity is always attractive—there is no 'gravitational repulsion.' Second, the equipotential surfaces are perpendicular to the field lines everywhere. Third, the spacing of field lines tells you about field strength: close together means a strong field, far apart means a weak field. Near the Earth's surface, the field lines are approximately parallel and equally spaced, which is why we treat g ≈ 9.81 N kg⁻¹ as roughly constant for everyday problems.

SECTION 4

Mathematical Framework

The mathematical backbone of gravitational fields rests on a few elegant equations. Each one connects a physical quantity—force, field strength, potential, or orbital speed—to the masses involved and the distance between them. Let's unpack the key formulas you need for IB Physics.

NEWTON'S LAW OF UNIVERSAL GRAVITATION

F = G m₁ m₂ / r²

F = gravitational force (N), G = 6.674 × 10⁻¹¹ N m² kg⁻² (universal gravitational constant), m₁ and m₂ = masses (kg), r = centre-to-centre distance (m).

GRAVITATIONAL FIELD STRENGTH

g = GM / r²

This gives the field strength at distance r from the centre of a mass M. Units: N kg⁻¹ or equivalently m s⁻². At Earth's surface (r ≈ 6.37 × 10⁶ m), this yields g ≈ 9.81 m s⁻².

GRAVITATIONAL POTENTIAL

V = −GM / r

V = gravitational potential (J kg⁻¹). The negative sign means work must be done against gravity to move a mass from that point to infinity. At infinity, V = 0.

ORBITAL SPEED (CIRCULAR ORBIT)

v = √(GM / r)

Derived by setting the gravitational force equal to the centripetal force: GMm / r² = mv² / r. The mass m of the orbiting body cancels out, so orbital speed depends only on M and r.

💡 IB Exam Tip

Always use the centre-to-centre distance for r, not the surface-to-surface distance. For a satellite orbiting at altitude h above Earth's surface, use r = R_E + h, where R_E is Earth's radius.

SECTION 5

How Gravitational Field Strength Varies with Distance

One of the most important skills in IB Physics is understanding how g changes as you move away from a planet or star. The relationship g = GM / r² is an inverse-square law: if you double the distance from a planet's centre, the field strength drops to one-quarter. The graph below illustrates this behaviour, showing how g falls off outside a uniform sphere and remains approximately constant inside the sphere (a result you may encounter at higher level).

The graph shows gravitational field strength g as a function of distance r from the centre of a uniform sphere. Field strength peaks at the surface (r = R) and follows an inverse-square decrease beyond. At 2R, g has dropped to g₀/4; at 3R, it is g₀/9.

Inverse-square drop-off of g outside a uniform sphere

Distance from Centre	Multiple of R	g as fraction of g₀
At the surface	1R	g₀
Double the radius	2R	g₀ / 4
Triple the radius	3R	g₀ / 9
Ten times the radius	10R	g₀ / 100

SECTION 6

Worked Example — Satellite in Orbit

Let's put all the equations together with a realistic IB-style problem. The International Space Station (ISS) orbits Earth at an altitude of approximately 408 km. We will calculate its orbital speed, the gravitational field strength at its altitude, and its orbital period.

📋 Given Data

Mass of Earth: M_E = 5.97 × 10²⁴ kg. Radius of Earth: R_E = 6.37 × 10⁶ m. Altitude: h = 408 km = 4.08 × 10⁵ m. G = 6.674 × 10⁻¹¹ N m² kg⁻².

ISS Orbital Calculations

Step 1 — Find the orbital radiusThe orbital radius is measured from Earth's centre, not from the surface. So r = R_E + h = 6.37 × 10⁶ + 4.08 × 10⁵ = 6.778 × 10⁶ m.

r = 6.778 × 10⁶ m

Step 2 — Calculate gravitational field strength at ISS altitudeUsing g = GM / r²: g = (6.674 × 10⁻¹¹ × 5.97 × 10²⁴) / (6.778 × 10⁶)². First compute the numerator: 6.674 × 10⁻¹¹ × 5.97 × 10²⁴ = 3.984 × 10¹⁴. Then the denominator: (6.778 × 10⁶)² = 4.594 × 10¹³. So g = 3.984 × 10¹⁴ / 4.594 × 10¹³ ≈ 8.67 N kg⁻¹.

g ≈ 8.67 N kg⁻¹ (about 88% of surface gravity — astronauts are NOT weightless because gravity vanishes!)

Step 3 — Calculate orbital speedUsing v = √(GM / r): v = √(3.984 × 10¹⁴ / 6.778 × 10⁶) = √(5.878 × 10⁷) ≈ 7668 m s⁻¹. That's about 27 600 km h⁻¹!

v ≈ 7.67 × 10³ m s⁻¹

Step 4 — Calculate orbital periodThe circumference of the orbit is C = 2πr = 2π × 6.778 × 10⁶ = 4.259 × 10⁷ m. The period T = C / v = 4.259 × 10⁷ / 7668 ≈ 5554 s. Converting to minutes: 5554 / 60 ≈ 92.6 minutes.

T ≈ 5550 s ≈ 92.6 min — the ISS completes roughly 15.5 orbits per day.

🛰️ WHY ARE ASTRONAUTS 'WEIGHTLESS'?

As Step 2 shows, the gravitational field at ISS altitude is about 8.67 N kg⁻¹ — only 12% less than on the ground. Astronauts feel weightless not because gravity has disappeared, but because they and the ISS are in free fall together. Imagine standing in a lift whose cable snaps — you and the floor fall at the same rate, so you'd feel weightless. The ISS is just 'falling around' the Earth rather than into it.

SECTION 7

Strengths & Limitations of Newton's Gravitational Model

Newton's model of gravitation is astonishingly powerful—it sent humans to the Moon and still guides spacecraft today. But like any model, it has limits. Understanding where it works well and where it breaks down is an important part of physics.

Comparison of Newtonian gravity's capabilities and its boundaries

Strengths	Limitations
Accurately predicts planetary orbits, satellite trajectories, and tidal forces.	Cannot explain the precession of Mercury's perihelion (43 arcseconds per century off).
Uses simple algebra—accessible without calculus for most IB problems.	Treats gravity as an instantaneous force with no propagation delay.
Works excellently for weak gravitational fields and speeds much less than the speed of light.	Breaks down near extremely massive objects like black holes or neutron stars.
The gravitational constant G has been measured to high precision (Cavendish experiment).	Does not predict gravitational lensing (bending of light by gravity), which general relativity does.

🔧 THE RIGHT TOOL FOR THE JOB

Newton's model is like a reliable calculator: it gives the right answer for nearly every IB-level scenario you will encounter. General relativity is the supercomputer you bring in only when dealing with extreme conditions—near black holes, at relativistic speeds, or when GPS satellites need nanosecond precision. For your exams, Newton's equations are the ones to master.

SECTION 8

Connection to General Relativity & Advanced Theory

While IB Physics D.1 focuses on Newtonian gravitation, it is valuable to see how this framework connects to Einstein's general theory of relativity. In Einstein's picture, mass and energy curve the fabric of spacetime itself, and objects follow the straightest possible paths (called geodesics) through that curved spacetime. What Newton calls a 'force,' Einstein explains as geometry.

Newtonian Gravity vs General Relativity

Feature	Newton's Gravity	Einstein's General Relativity
Nature of gravity	A force between masses	Curvature of spacetime
Speed of propagation	Instantaneous	Speed of light (gravitational waves)
Effect on light	Not predicted	Predicts gravitational lensing
Effect on time	Not addressed	Predicts gravitational time dilation
Accuracy for everyday orbits	Excellent	Excellent (tiny corrections)

For the IB exam, you will not need to perform calculations using general relativity. However, examiners do expect you to recognise that Newton's model is a limiting case of a more complete theory. When you see questions about the limitations of the Newtonian model or the significance of Einstein's work, point to phenomena like Mercury's orbital precession, gravitational lensing, and gravitational time dilation as evidence that general relativity provides a deeper description of gravity.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Explain why gravitational potential is always negative, and state the value of gravitational potential at an infinite distance from any mass.

PROBLEM 2 — BASIC CALCULATION

Calculate the gravitational field strength at the surface of Mars. Use M_Mars = 6.42 × 10²³ kg and R_Mars = 3.39 × 10⁶ m.

PROBLEM 3 — INTERMEDIATE

A geostationary satellite orbits Earth with a period of exactly 24.0 hours. Show that its orbital radius is approximately 4.22 × 10⁷ m. (Hint: equate gravitational force to centripetal force and use T = 2πr / v.)

PROBLEM 4 — APPLIED

The Hubble Space Telescope orbits at an altitude of 547 km above Earth's surface. (a) Determine the gravitational potential at Hubble's orbit. (b) Calculate the energy required to move a 1.0 kg mass from Earth's surface to Hubble's altitude. Use M_E = 5.97 × 10²⁴ kg and R_E = 6.37 × 10⁶ m.

PROBLEM 5 — CRITICAL THINKING

Two identical spheres, each of mass M, are separated by a distance d (centre to centre). (a) Determine the location between the two spheres where the net gravitational field strength is zero. (b) Is the gravitational potential also zero at this point? Justify your answer.

SUMMARY

Lesson Summary

In this lesson you explored how Newton's law of universal gravitation (F = Gm₁m₂ / r²) provides the foundation for understanding gravitational fields. You learned that gravitational field strength g = GM / r² follows an inverse-square law, and that gravitational potential V = −GM / r is always negative because gravity is always attractive. Field lines point radially inward and are perpendicular to equipotential surfaces.

You applied these principles to calculate orbital speed (v = √(GM / r)), orbital period, and field strength at altitude. You also examined the strengths and limitations of Newton's model, noting that while it excels for everyday and IB-level problems, general relativity provides a more complete description at extreme scales. Remember: the principle of superposition lets you add gravitational field vectors from multiple masses, and always use the centre-to-centre distance for r in your calculations.