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Gravitational Force and Field

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Why this chapter
Why this chapter exists

The force that makes a mango fall also holds the Moon in its path and the planets round the Sun. Newton wrote it as one short law, and this chapter builds everything else on it. First the law itself and the way many pulls add. Then the field, which describes the pull at every point before any body is placed there. Rods, rings and shells show how an extended body pulls, and the shell theorems let every planet act as a point. Last comes g itself. We find why it is 9.8 m/s², and why it changes a little on a mountain, down a mine and at the equator.

You will learn
By the end of this chapter you will
  1. State Newton's law of gravitation in words, as a formula and as a vector, and use its inverse-square dependence.
  2. Find the net gravitational force on a body from several masses by superposition, using signs on a line and symmetry in a plane.
  3. Define the gravitational field as force per unit mass, find the field of a point mass, and locate the null point between two masses.
  4. Derive the fields of a uniform rod and a ring, and find where the ring's field on its axis is greatest.
  5. State and use the shell theorems for a hollow shell and a uniform solid sphere, inside and outside.
  6. Derive \(g = GM/R^2\), and find how \(g\) changes with height, depth, latitude and the Earth's shape.
Before you begin
Before you begin

From the chapter on Forces and Newton's Laws, you should know the second law and third-law pairs. You should be able to add vectors by components. From the chapters on Circular Motion, you should know centripetal acceleration and the centrifugal force in a rotating frame. From the chapter on Centre of Mass, you should be able to cut a body into small elements and add them by an integral. In this unit we take \(g = 9.8\,\text{m/s}^2\), \(G = 6.67 \times 10^{-11}\,\text{N m}^2/\text{kg}^2\) and \(R_E = 6.4 \times 10^6\,\text{m}\).

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Lessons

1
Newton's Law of Gravitation
Every particle attracts every other particle. The force grows with the product of the masses and falls as the square of the distance. One tiny constant, G, sets its size, and the same law moves a falling mango and the Moon.
The law of universal gravitationProperties of the gravitational forceThe gravitational constantThe moon falls like the mango
15 min0%
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2
Superposition of Gravitational Forces
When many masses pull on one body, each pull is found as if the others were absent, and the pulls add as vectors. On a line that is a sum with signs; in a plane it needs components, and symmetry often does most of the work.
The principle of superpositionNet force on a mass on a lineForces in a plane and symmetry
13 min0%
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3
The Gravitational Field
The pull on a body depends on its mass, but the pull per kilogram belongs to the place. That is the gravitational field. A point mass makes a field that points inward and falls as \(1/r^2\). Two masses make a point where their fields cancel. Near the ground the field is uniform.
Field is force per unit massField of a point massThe null point between two massesThe uniform field near the ground
15 min0%
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4
Fields of Rods, Rings and Shells
An extended body is a sum of small masses, so its field is a sum, or an integral, of point-mass fields. The rod, the ring and the spherical shell are the three standard cases, and the shell theorems let every uniform sphere act as a point.
Field of a uniform rodField on the axis of a ringThe shell theorems
16 min0%
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5
The Acceleration Due to Gravity and How It Varies
\(g\) is the Earth's field at its surface, \(GM/R^2\). It falls as you climb and as you go down a tunnel. It is smaller at the equator than at the poles, because the Earth spins and bulges.
G from the law of gravitationG above the surfaceG below the surfaceG with latitude and the earths shape
16 min0%
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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

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The Workbook
Questions to solve on paper, marked by you.
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Glossary

The key words of this chapter, with the meaning this chapter gives them.

acceleration due to gravity
The acceleration of a body falling freely near a planet, equal to the planet's field there. At the Earth's surface \(g = GM/R^2 = 9.8\,\text{m/s}^2\).
central force
A force that acts along the line joining two bodies, with a size that depends only on their distance. Gravitation is a central force.
effective g
The acceleration due to gravity that a body at rest on the spinning Earth actually shows, \(g' = g - \omega^2R\cos^2\lambda\) at latitude \(\lambda\).
equatorial bulge
The slight swelling of the Earth at the equator caused by its spin. The equatorial radius is about \(21\,\text{km}\) larger than the polar radius.
field line
A line drawn so that its direction at every point is the direction of the field there. Gravitational field lines point towards the masses and crowd where the field is strong.
gravitation
The attraction that every mass exerts on every other mass. Weight is the gravitational pull of the Earth on a body near it.
gravitational constant
The universal constant \(G = 6.67 \times 10^{-11}\,\text{N m}^2/\text{kg}^2\) in Newton's law of gravitation. It is the force between two \(1\,\text{kg}\) masses \(1\,\text{m}\) apart.
gravitational field
The gravitational force per unit mass on a small test mass at a point, \(\vec E = \vec F/m\). It is measured in N/kg, the same as m/s².
inverse-square law
A law in which a quantity falls as the square of the distance, as \(1/r^2\). Twice the distance gives a quarter of the value.
latitude
The angle between the equator's plane and the line from the Earth's centre to a point on the surface. It is \(0°\) at the equator and \(90°\) at a pole.
linear mass density
The mass per unit length of a thin rod or wire, \(\lambda = M/L\), in kg/m.
null point
A point where the gravitational fields of two or more masses cancel, so that a body placed there feels no net gravitational force.
point mass
A body treated as a single point that has mass but no size. Newton's law of gravitation is written for point masses.
principle of superposition
The rule that the net gravitational force on a body is the vector sum of the separate pulls. Each pull is found as if the other masses were absent.
shell theorem
A uniform spherical shell attracts an outside point as if all its mass were at its centre. It exerts no net force on any point inside it.
spherical shell
A thin hollow sphere of uniform thickness and density.
test mass
A mass small enough that its own pull does not disturb the masses whose field it is used to measure.
uniform field
A field that has the same size and the same direction at every point of a region. Near the ground the Earth's field is uniform to a very good approximation.