Understand Gravitational Fields — Understand D.1 Gravitational fields - IB Physics

SECTION 1

Historical Context & Motivation

Humans have always wondered why objects fall to the ground, why the Moon circles the Earth, and why planets trace predictable paths across the night sky. For centuries, thinkers proposed different explanations—some mystical, some mechanical—but none could unify terrestrial and celestial motion into a single framework. The concept of a gravitational field eventually emerged as the key idea that ties all of these phenomena together, describing how mass warps the space around it and influences other masses at a distance.

~350 BCE • Aristotle's Natural Motion

Aristotle taught that heavy objects have a natural tendency to move toward the centre of the Earth. While incorrect in mechanism, his ideas dominated European thought for nearly two thousand years.

1687 • Newton's Law of Universal Gravitation

Isaac Newton published the Principia, showing that the same force causing an apple to fall also keeps the Moon in orbit. He proposed that every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

1798 • Cavendish Measures G

Henry Cavendish used a sensitive torsion balance to measure the gravitational constant G, effectively 'weighing' the Earth. His experiment confirmed Newton's law quantitatively and gave science a precise value for the strength of gravity.

1915 • Einstein's General Relativity

Albert Einstein reinterpreted gravity not as a force but as the curvature of spacetime caused by mass and energy. For most IB-level problems, Newton's description remains perfectly accurate, but Einstein's theory explains extreme situations like black holes and gravitational waves.

The central question that motivates this topic is: How can we describe the gravitational influence of a mass at every point in space, even before another object arrives to 'feel' it? Answering this question leads us to the field concept—a powerful tool that lets us map gravitational effects around any mass, predict orbital motion, and calculate escape speeds.

SECTION 2

Core Principles & Definitions

A gravitational field is a region of space in which a mass experiences a gravitational force. Instead of thinking about forces acting mysteriously across empty space, we say that any object with mass creates a field around itself, and any other mass placed in that field feels a force. This shifts our perspective from 'action at a distance' to something more tangible: the field exists whether or not a second mass is present to detect it.

Gravitational Field Strength (g)

The gravitational field strength at a point is defined as the gravitational force per unit mass experienced by a small test mass placed at that point. Its SI unit is N kg⁻¹ (equivalent to m s⁻²). Near Earth's surface, g ≈ 9.81 N kg⁻¹.

Field Lines

Gravitational field lines are imaginary lines that show the direction of the gravitational force on a test mass. They always point toward the mass creating the field (because gravity is always attractive) and their spacing indicates field strength—closer lines mean a stronger field.

Uniform vs. Radial Fields

A uniform field has parallel, equally spaced field lines (approximately true near Earth's surface over small regions). A radial field has field lines converging toward a point mass, with strength decreasing as distance increases.

Newton's Law of Universal Gravitation

Every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the separation. The constant of proportionality is G = 6.674 × 10⁻¹¹ N m² kg⁻².

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KEY TAKEAWAY

Think of a gravitational field like the warmth around a campfire. The fire radiates heat whether or not anyone is standing nearby. Move closer and you feel more heat; step back and you feel less. Similarly, a mass 'radiates' a gravitational field in every direction—any other mass placed in that region will feel a pull, and the closer it is, the stronger the pull.

SECTION 3

Visualising Gravitational Fields

Diagrams are essential for understanding gravitational fields. The following diagram shows two scenarios side by side: a radial field around an isolated point mass (or spherical body like Earth) and a uniform field near the surface of a planet over a small region. Notice how field lines in the radial diagram converge toward the centre and grow farther apart as you move away, while the uniform field lines are parallel and equally spaced.

Left: A radial gravitational field around a spherical mass M. Arrows point inward because gravity is always attractive. Right: A uniform gravitational field near the surface of a planet. The parallel, equally spaced arrows indicate that g is approximately constant over this small region.

In the radial field diagram, notice that the arrows all point inward—toward the mass. This is a distinguishing feature of gravitational fields: unlike electric fields, which can be attractive or repulsive, gravitational fields are always attractive. The spacing between lines tells you about the field strength at different distances. Close to the mass, lines are bunched together, meaning g is large. Far away, lines spread apart and g is weaker. In the uniform field approximation on the right, we treat the surface as flat and the field lines as perfectly parallel, which works well for problems involving projectile motion and free fall near Earth's surface.

SECTION 4

Mathematical Framework

The mathematics of gravitational fields follows directly from Newton's law. We begin with the force law itself, then derive the gravitational field strength formula. These two equations are the workhorses of IB gravitational field problems.

NEWTON'S LAW OF UNIVERSAL GRAVITATION

F = G × M × m / r²

F = gravitational force between two masses (N); G = universal gravitational constant = 6.674 × 10⁻¹¹ N m² kg⁻²; M = mass of the field-creating body (kg); m = mass of the object experiencing the force (kg); r = distance between the centres of the two masses (m).

The gravitational field strength at a point is defined as force per unit mass. If we place a small test mass m in the field of a larger mass M, we can divide both sides of Newton's law by m to isolate the field strength.

GRAVITATIONAL FIELD STRENGTH

g = F / m = G × M / r²

g = gravitational field strength at distance r from mass M (N kg⁻¹ or m s⁻²). Note that g depends only on the field-creating mass M and the distance r—it does not depend on the test mass.

GRAVITATIONAL POTENTIAL ENERGY

E_p = −G × M × m / r

E_p = gravitational potential energy (J). The negative sign indicates that work must be done against gravity to move the mass m to infinity. At r = ∞, E_p = 0 by convention.

GRAVITATIONAL POTENTIAL

V_g = −G × M / r

V_g = gravitational potential at distance r from mass M (J kg⁻¹). This is the energy per unit mass required to move a test mass from that point to infinity. It is always negative because gravity is attractive.

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IB Exam Tip

Remember that r in these equations is always the distance from the centre of the mass, not from its surface. If a problem gives you the altitude above a planet's surface, you must add the planet's radius to get r.

SECTION 5

Field Patterns & the Inverse-Square Law

The inverse-square law is central to understanding how gravitational field strength changes with distance. Because g ∝ 1/r², doubling the distance from the centre of a mass reduces the field strength to one quarter of its original value. Tripling the distance reduces it to one ninth. This rapid decrease explains why gravity from the Sun is strong enough to hold Earth in orbit but far too weak for us to feel on our skin.

The graph shows how gravitational field strength g falls off with distance r from the centre of a uniform spherical mass. At the surface (r = R), g has its maximum value g₀. At twice the radius (r = 2R), g drops to g₀/4. The amber curve follows the inverse-square relationship and asymptotically approaches zero but never quite reaches it.

The graph above is one of the most commonly examined visuals in IB Physics. Notice that the curve has a steep initial drop and then flattens out gradually. Gravitational field strength theoretically extends to infinity but becomes negligibly small at large distances. Inside the planet (the shaded blue region), the relationship changes: g increases approximately linearly from zero at the centre to its surface value, because only the mass enclosed within your radius contributes to the field. For IB D.1, you primarily deal with the region outside the mass (r ≥ R).

Distance from centre	Field strength g	Fraction of surface value
r = R (surface)	g₀	1 (100%)
r = 2R	g₀ / 4	1/4 (25%)
r = 3R	g₀ / 9	1/9 (≈ 11%)
r = 4R	g₀ / 16	1/16 (≈ 6.3%)
r = 10R	g₀ / 100	1/100 (1%)

How g decreases with distance for a spherical mass

SECTION 6

Worked Example

Let's work through a typical IB-style problem that combines Newton's law of gravitation with the concept of gravitational field strength.

Finding gravitational field strength at the altitude of the ISS

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Step 1 — Identify Given Values

The International Space Station (ISS) orbits at an altitude of approximately 408 km above Earth's surface. We need: mass of Earth, M = 5.97 × 10²⁴ kg; radius of Earth, R = 6.371 × 10⁶ m; altitude h = 408 × 10³ m = 4.08 × 10⁵ m; G = 6.674 × 10⁻¹¹ N m² kg⁻².
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Step 2 — Calculate distance from Earth's centre

The formula requires r measured from the centre of the Earth, not from the surface. So we add the planet's radius to the altitude: r = R + h = 6.371 × 10⁶ + 4.08 × 10⁵ = 6.779 × 10⁶ m.

r = 6.779 × 10⁶ m
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Step 3 — Substitute into g = GM/r²

g = (6.674 × 10⁻¹¹ × 5.97 × 10²⁴) / (6.779 × 10⁶)²
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Step 4 — Evaluate the numerator

Numerator = 6.674 × 10⁻¹¹ × 5.97 × 10²⁴ = 3.984 × 10¹⁴ N m² kg⁻¹.
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Step 5 — Evaluate the denominator and divide

Denominator = (6.779 × 10⁶)² = 4.595 × 10¹³ m². Therefore g = 3.984 × 10¹⁴ / 4.595 × 10¹³ ≈ 8.67 N kg⁻¹.

g ≈ 8.67 N kg⁻¹
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Step 6 — Interpret the result

At the ISS altitude, g is about 88% of its surface value (9.81 N kg⁻¹). Astronauts are not truly in 'zero gravity'—they are in free fall. The gravitational field is still very strong; they just happen to be falling around the Earth at the same rate as the station.

SECTION 7

Gravitational vs. Electric Fields

IB Physics treats gravitational and electric fields together under the Fields topic. Understanding their similarities and differences will deepen your grasp of the field concept and help you transfer knowledge from one topic to the other. The table below highlights the key comparisons.

Property	Gravitational Field	Electric Field
Source	Mass	Electric charge
Nature	Always attractive	Attractive or repulsive
Field strength formula	g = GM/r²	E = kQ/r²
Potential formula	V = −GM/r (always negative)	V = kQ/r (sign depends on charge)
Distance dependence	Inverse-square (1/r²)	Inverse-square (1/r²)
Shielding possible?	No — cannot be blocked	Yes — conductors can shield
Relative strength	Extremely weak	Much stronger (≈ 10³⁶ times)

Comparison of gravitational and electric fields

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KEY TAKEAWAY

Gravitational and electric fields share the same mathematical structure—both follow inverse-square laws and can be described using field lines, field strength, and potential. The critical difference is that gravity is always attractive and cannot be shielded, while electric forces can be attractive or repulsive and can be blocked by conductors. Learning one field type gives you a head start on the other.

SECTION 8

Connection to General Relativity

Newton's gravitational field model works brilliantly for almost every scenario you will encounter in IB Physics—and indeed for most engineering applications. However, it breaks down in extreme situations: near black holes, at speeds close to the speed of light, or when measuring tiny effects like the precession of Mercury's orbit. Einstein's general theory of relativity (1915) reinterprets gravity not as a force transmitted through a field, but as the curvature of spacetime itself. Masses cause spacetime to warp, and other objects follow curved paths (geodesics) through that warped spacetime.

Feature	Newtonian Gravity	General Relativity
Nature of gravity	Force between masses	Curvature of spacetime
Speed of propagation	Instantaneous (assumed)	Speed of light, c
Works well for…	Everyday masses, moderate speeds, weak fields	All scenarios including extreme mass & speed
Mathematical difficulty	Algebra and basic calculus	Tensor calculus (university-level)
IB Exam relevance	Core tool — used in calculations	Conceptual awareness only

Newtonian gravity vs. General Relativity

For the IB exam, you should know that general relativity exists and that it predicts phenomena Newton's model cannot explain—such as gravitational lensing (the bending of light around massive objects), gravitational time dilation (clocks running slower in stronger gravitational fields), and gravitational waves (ripples in spacetime detected for the first time in 2015 by LIGO). However, all numerical calculations in IB D.1 use Newton's equations.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why astronauts aboard the International Space Station appear to float even though the gravitational field strength at their altitude is approximately 8.7 N kg⁻¹. Why is the term 'zero gravity' misleading?

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The astronauts and the ISS are both in free fall—they are continuously falling toward Earth but moving sideways fast enough that they keep missing it. Because everything inside the station falls at the same rate, there is no normal contact force between an astronaut and the floor, which creates the sensation of weightlessness. The gravitational field at 408 km altitude is still about 88% of the surface value, so 'zero gravity' is inaccurate. A better term is 'microgravity' or simply 'free fall.'

Question 2: BASIC CALCULATION

Calculate the gravitational field strength on the surface of Mars. Data: mass of Mars = 6.42 × 10²³ kg, radius of Mars = 3.39 × 10⁶ m.

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Using g = GM/r²: g = (6.674 × 10⁻¹¹ × 6.42 × 10²³) / (3.39 × 10⁶)². The numerator equals 4.285 × 10¹³ and the denominator equals 1.150 × 10¹³. Dividing gives g ≈ 3.73 N kg⁻¹ (or m s⁻²). This is roughly 38% of Earth's surface gravity, which is why Mars rovers can be lighter and why future human explorers would feel noticeably lighter when walking on the Martian surface.

Question 3: INTERMEDIATE

A satellite orbits Earth at an altitude where the gravitational field strength is exactly one quarter of the surface value (g_surface = 9.81 N kg⁻¹). Calculate the satellite's altitude above Earth's surface. (R_E = 6.371 × 10⁶ m.)

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If g = g₀/4, then since g ∝ 1/r², we need r² to be 4 times larger than R², meaning r = 2R. So the distance from Earth's centre is r = 2 × 6.371 × 10⁶ = 1.274 × 10⁷ m. The altitude above the surface is h = r − R = 1.274 × 10⁷ − 6.371 × 10⁶ = 6.371 × 10⁶ m, which is about 6370 km. Notice that the satellite is at an altitude equal to one Earth radius—this neatly illustrates the inverse-square law: doubling r from the centre quarters the field strength.

Question 4: APPLIED

The gravitational potential at the surface of a newly discovered exoplanet is −4.5 × 10⁷ J kg⁻¹ and its radius is 7.2 × 10⁶ m. Determine the mass of the exoplanet and the gravitational field strength at its surface.

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From V = −GM/r, we can solve for M: M = −Vr/G = (4.5 × 10⁷ × 7.2 × 10⁶) / 6.674 × 10⁻¹¹. The numerator is 3.24 × 10¹⁴ and dividing by G gives M ≈ 4.86 × 10²⁴ kg—roughly 81% of Earth's mass. For the field strength, g = GM/r² = (6.674 × 10⁻¹¹ × 4.86 × 10²⁴) / (7.2 × 10⁶)² = 3.243 × 10¹⁴ / 5.184 × 10¹³ ≈ 6.25 N kg⁻¹. Alternatively, g = −V/r = 4.5 × 10⁷ / 7.2 × 10⁶ = 6.25 N kg⁻¹, which provides a useful cross-check.

Question 5: CRITICAL THINKING

Two identical spherical masses, each of mass M, are separated by a distance d (centre to centre). (a) At what point along the line joining their centres is the net gravitational field strength zero? (b) Is this point a stable or unstable equilibrium position for a small test mass? Justify your answer.

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(a) By symmetry, the net field is zero at the exact midpoint between the two masses, at a distance d/2 from each centre. At this point, the gravitational field due to the left mass pulls the test mass to the left with the same magnitude that the right mass pulls it to the right, so the vector sum is zero. (b) This is an unstable equilibrium. If the test mass is displaced slightly toward one of the masses, the field from the nearer mass increases (inverse-square law) while the field from the farther mass decreases. The net force pushes the test mass further from the midpoint rather than restoring it. This is analogous to balancing a ball on top of a hill—any slight push sends it rolling away. In gravitational potential terms, the midpoint is a local maximum of potential (least negative), not a minimum, confirming instability.

SUMMARY

Lesson Summary

Summary

A gravitational field is a region of space where a mass experiences a gravitational force. The gravitational field strength g = GM/r² tells us the force per unit mass at any point and has units of N kg⁻¹. Newton's law of universal gravitation (F = GMm/r²) describes the attractive force between any two masses. The inverse-square law means that doubling the distance from a mass quarters the field strength. Near a planet's surface, the field is approximately uniform (parallel, equally spaced field lines), while farther away it is radial (lines converging toward the centre).

Gravitational potential V = −GM/r is always negative (gravity is always attractive) and represents the energy per unit mass needed to move a test mass to infinity. The gravitational potential energy $E_p$ = −GMm/r follows the same pattern. Remember that r is always measured from the centre of the mass. Gravitational fields share the same mathematical structure as electric fields but are exclusively attractive and cannot be shielded. For extreme scenarios, Einstein's general relativity replaces Newton's model, but Newtonian gravity remains the primary calculation tool for IB D.1.