Gravitational field

1. Overview

The concept of a gravitational field is fundamental to understanding how objects with mass interact across space without direct physical contact. In classical physics, this is described as a field of force: a region where a mass experiences a non-contact force. Rather than viewing gravity as an instantaneous "pull" between two distant objects, field theory suggests that any object with mass alters the properties of the space surrounding it. This "alteration" is the gravitational field. When another mass enters this region, it interacts with the field locally, experiencing a force.

This topic establishes the quantitative framework for measuring the intensity of these fields through gravitational field strength ($g$) and provides the qualitative visual language of field lines to map the direction and magnitude of gravitational influences. Mastery of these concepts is the prerequisite for understanding orbital mechanics, planetary motion, and gravitational potential in subsequent sections of the Cambridge 9702 syllabus.

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Key Definitions

• Field of Force: A region of space where a specific type of object (such as a mass, a charge, or a magnet) experiences a force.
• Gravitational Field: A region of space where a mass experiences a gravitational force.
• Gravitational Field Strength ($g$): The gravitational force per unit mass acting on a small point mass placed at a specific point in the field.
• Vector Quantity: A quantity that possesses both magnitude and direction. Gravitational field strength is a vector because it is derived from force.
• Point Mass: An idealized model of an object where all its mass is considered to be concentrated at a single point in space.
• Radial Field: A field produced by a point mass or a uniform spherical mass, where field lines converge toward the center of mass.
• Uniform Field: A field in which the gravitational field strength is constant in both magnitude and direction at all points.
• Test Mass: A mass small enough that its own gravitational field does not significantly perturb or alter the field being measured.

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Content

3.1 The Nature of Gravitational Fields

A gravitational field is a vector field. This means that every point in the space surrounding a mass has a specific value of field strength associated with it, and that value has a defined direction.

1. Source of the Field: Every object with mass ($M$) creates a gravitational field. The field exists regardless of whether another mass is present to "feel" it.
2. Interaction: When a second mass ($m$) is placed in the field of $M$, the field exerts an attractive force on $m$. According to Newton’s Third Law, $m$ also creates its own field which exerts an equal and opposite force on $M$.
3. Range: Theoretically, the gravitational field of any mass extends to infinity, although its strength diminishes rapidly with distance.

3.2 Defining Gravitational Field Strength ($g$)

The gravitational field strength at a point is defined quantitatively as the ratio of the gravitational force to the mass of the object experiencing that force.

The Fundamental Equation:

$$\mathbf{g = \frac{F}{m}}$$

Where:

• $g$ is the gravitational field strength, measured in Newtons per kilogram ($\text{N kg}^{-1}$).
• $F$ is the gravitational force acting on the mass, measured in Newtons ($\text{N}$).
• $m$ is the mass of the object placed in the field (the "test mass"), measured in kilograms ($\text{kg}$).

Key Conceptual Details:

• The "Unit Mass" Requirement: In definitions, the term "unit mass" is a mathematical way of saying $1\text{ kg}$. By defining $g$ as force per unit mass, we create a value that is independent of the size of the test mass used to measure it.
• The "Small Point Mass" Requirement: We specify a "small" mass to ensure that the test mass's own gravity doesn't move the source mass or significantly change the field configuration. We specify "point mass" so that the measurement is taken at a precise location in space.
• Direction: The direction of the vector $g$ is always the direction of the gravitational force. Since gravity is always attractive, $g$ always points towards the center of the mass creating the field.
• Units and Dimensions: While $\text{N kg}^{-1}$ is the standard unit for field strength, it is dimensionally identical to $\text{m s}^{-2}$ (acceleration). Thus, $g$ also represents the acceleration of free fall. In the context of fields, $\text{N kg}^{-1}$ is preferred to emphasize the "force per unit mass" nature.

3.3 Representation by Field Lines

Gravitational field lines (also called lines of force) are a geometric representation used to visualize the field's structure. They follow strict rules:

1. Direction of Arrows: Arrows must be drawn on the lines to show the direction of the force on a test mass. For gravity, these arrows always point towards the source mass.
2. Line Density (Flux Density): The spacing of the lines indicates the magnitude of $g$.
   • Close lines = Strong field.
   • Widely spaced lines = Weak field.
3. Perpendicularity: Field lines must meet the surface of a spherical mass or a boundary perpendicularly ($90^\circ$).
4. Continuity: Field lines do not start or stop in empty space; they originate at infinity and terminate at the mass. They never cross or form closed loops.

3.4 Types of Gravitational Fields

A. Radial Fields A radial field is produced by a point mass or a perfectly uniform sphere (like a planet or star).

• Geometry: The lines resemble the spokes of a wheel, all directed toward the center of the mass.
• Variation: As you move further from the center, the lines spread out. This visual spreading represents the inverse square law—the field strength $g$ decreases as the distance $r$ increases ($g \propto 1/r^2$).
• Mathematical Note: At the surface of a planet of radius $R$, the field is strongest. As $r \to \infty$, $g \to 0$.

B. Uniform Fields A uniform field is a simplification used for small-scale regions.

• Geometry: Represented by parallel, equally spaced straight lines.
• Context: Near the surface of a large planet (like Earth), the curvature of the surface is so slight that, over a few kilometers, the field lines appear parallel and the value of $g$ remains effectively constant (e.g., $9.81\text{ N kg}^{-1}$).
• Criteria: For a field to be truly uniform, $g$ must have the same magnitude and the same direction at every point in the region.

C. Resultant Fields and Null Points When multiple masses are present, the total gravitational field at any point is the vector sum of the individual fields.

• Null Point: A point in space where the gravitational field strength from one mass is exactly equal in magnitude and opposite in direction to the field strength from another mass. The resultant $g$ is zero.
• Example: Between the Earth and the Moon, there is a point where a spacecraft would experience no net gravitational force.

3.5 Worked Examples

Worked Example 1 — Calculating Field Strength from Force

A probe with a mass of $450\text{ kg}$ is positioned in space where it experiences a gravitational pull of $1350\text{ N}$ from a nearby planet. Calculate the gravitational field strength at that location.

• Step 1: State the known values. $m = 450\text{ kg}$ $F = 1350\text{ N}$
• Step 2: State the relevant equation. $$g = \frac{F}{m}$$
• Step 3: Substitute the values. $$g = \frac{1350\text{ N}}{450\text{ kg}}$$
• Step 4: Calculate and state the final answer with units. $$g = 3.0\text{ N kg}^{-1}$$

Worked Example 2 — Determining Force in a Planetary Field

The gravitational field strength on the surface of Mars is approximately $3.7\text{ N kg}^{-1}$. A robotic rover has a mass of $900\text{ kg}$ on Earth. Calculate the weight (gravitational force) of the rover when it is deployed on the Martian surface.

• Step 1: Identify that mass is constant. The mass $m$ remains $900\text{ kg}$ regardless of location.
• Step 2: Rearrange the field strength formula. $$F = m \cdot g$$
• Step 3: Substitute the Martian field strength. $$F = 900\text{ kg} \times 3.7\text{ N kg}^{-1}$$
• Step 4: Final calculation. $$F = 3330\text{ N}$$ (Note: To 2 significant figures, $F = 3300\text{ N}$).

Worked Example 3 — Resultant Field Strength (Vector Addition)

An object is placed at a point exactly halfway between two identical stars, Star A and Star B. Each star creates a gravitational field of magnitude $g_s$ at that midpoint. Determine the resultant gravitational field strength at this point and describe the motion of a mass released there.

• Step 1: Analyze the directions. The field from Star A ($g_A$) points toward the center of Star A. The field from Star B ($g_B$) points toward the center of Star B.
• Step 2: Vector summation. Since the point is halfway and the stars are identical, $|g_A| = |g_B|$. Because they point in opposite directions: $g_{total} = g_A + (-g_B) = 0$.
• Step 3: Conclusion. The resultant field strength is $0\text{ N kg}^{-1}$. This is a null point.
• Step 4: Motion. A mass released at this point would experience no net force and would remain stationary (in equilibrium).

Worked Example 4 — Unit Conversion and Precision

A small test mass of $250\text{ mg}$ experiences a force of $2.45 \times 10^{-3}\text{ N}$ at the surface of a small asteroid. Calculate the gravitational field strength of the asteroid.

• Step 1: Convert mass to SI units (kg). $$m = 250\text{ mg} = 250 \times 10^{-3}\text{ g} = 250 \times 10^{-6}\text{ kg} = 2.5 \times 10^{-4}\text{ kg}$$
• Step 2: Use the formula. $$g = \frac{F}{m}$$
• Step 3: Substitution. $$g = \frac{2.45 \times 10^{-3}\text{ N}}{2.5 \times 10^{-4}\text{ kg}}$$
• Step 4: Final Answer. $$g = 9.8\text{ N kg}^{-1}$$

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Key Equations

Equation	Description	Status
$g = \frac{F}{m}$	Definition of gravitational field strength: Force ($F$) per unit mass ($m$).	Memorise
$W = mg$	Weight: The force ($W$) experienced by a mass ($m$) in a field ($g$).	Memorise

Note: While $g = \frac{F}{m}$ is the definition, you will often see it rearranged as $F = mg$ to calculate the weight of an object.

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Common Mistakes to Avoid

• ❌ Defining $g$ as "Gravity": "Gravity" is a general phenomenon. In an exam, you must use the specific term "Gravitational Field Strength".
• ❌ Missing "Unit Mass" in Definitions: If asked to define $g$, simply saying "force on a mass" is insufficient. You must state "force per unit mass" or "force acting on a mass of 1 kg".
• ❌ Incorrect Field Line Direction: Students often confuse gravitational and electric fields. Gravitational field lines never point away from a mass. They are always attractive (pointing towards the mass).
• ❌ Confusing $g$ and $G$: $g$ is the field strength (variable, units $\text{N kg}^{-1}$), while $G$ is the Universal Gravitational Constant ($6.67 \times 10^{-11}\text{ N m}^2\text{ kg}^{-2}$). They are not the same.
• ❌ Drawing Non-Uniform "Uniform" Fields: In diagrams of uniform fields, if the lines are not perfectly parallel or the spacing is inconsistent, you will lose marks. Use a ruler.
• ❌ Neglecting Units: Always check if the mass is given in grams ($\text{g}$) or milligrams ($\text{mg}$) and convert to kilograms ($\text{kg}$) before calculating $g$.

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Exam Tips

1. The "Small Point Mass" Keyword: When defining $g$, always include the phrase "acting on a small point mass". Mark schemes often require this to ensure you understand that the test mass shouldn't distort the field.
2. Field Line Precision:
   • For a radial field: Use a ruler to draw at least 8 lines. Ensure they are symmetrically distributed and would all intersect exactly at the center of the circle if extended backwards.
   • For a uniform field: Draw at least 3-4 parallel lines. Use a ruler to ensure the distance between Line 1 and Line 2 is identical to the distance between Line 2 and Line 3.
3. Vector Addition: If a question involves two masses, remember that $g$ is a vector. If you are calculating the field strength at a point between two planets, you must subtract the magnitudes because the field vectors point in opposite directions.
4. Significant Figures: Standard A-Level practice is to provide answers to the same number of significant figures as the least precise data point given in the question (usually 2 or 3 s.f.).
5. Interpreting Diagrams: If an exam question shows field lines that are getting closer together, you must state that the field strength is increasing. If the lines are parallel and equally spaced, you must state the field is uniform.
6. Weight vs. Mass: Remember that mass is a scalar and constant for an object, while weight ($F = mg$) is a vector and changes depending on the local gravitational field strength $g$.