In simple terms
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Gravitational force between point masses
Cambridge 9702 Paper 4 - Gravitational force between point masses (13.2). Senpai Corner diagram-backed pilot with premium structure and live visuals.
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13.2 Gravitational force between point masses.
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For a point outside a uniform sphere, the mass of the sphere may be considered to be a point mass at its centre.
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A uniform sphere is one where its mass is distributed evenly .
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The gravitational field lines around a uniform sphere are therefore identical to those around a point mass.
What this topic covers
The official Cambridge syllabus points this lesson works through.
- 13.2.1
Understand that, for a point outside a uniform sphere, the mass of the sphere may be considered to be a point mass at its centre
- 13.2.2
Recall and use Newton's law of gravitation for the force between two point masses
- 13.2.3
Analyse circular orbits in gravitational fields by relating the gravitational force to the centripetal acceleration it causes
- 13.2.4
Understand that a satellite in a geostationary orbit remains at the same point above the Earth's surface, with an orbital period of 24 hours, orbiting from west to east, directly above the Equator
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Key formulas
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Full topic notes
Formal explanation with the rigour you need for the exam.
Unpacking Newton's Law of Gravitation
Sir Isaac Newton's groundbreaking law provides a precise way to determine the strength of the attractive force between any two objects possessing mass. It states that the force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. This 'inverse square' relationship is a key feature of many physical laws.
13.2 Gravitational force between point masses.
For a point outside a uniform sphere, the mass of the sphere may be considered to be a point mass at its centre.
A uniform sphere is one where its mass is distributed evenly .
The gravitational field lines around a uniform sphere are therefore identical to those around a point mass.
An object can be regarded as point mass when a body covers a very large distance as compared to its size .
Radial fields are considered non-uniform fields.
The Concept of a Point Mass and Uniform Spheres
Newton's law is defined for 'point masses' - objects with mass but considered to have zero volume. While no real object is a true point mass, this is an excellent approximation when the distance between objects is much larger than their individual sizes, such as in astronomy. A crucial extension of this concept, also proven by Newton, is that a uniform spherical body (one with evenly distributed mass) exerts a gravitational force on an external object as if its entire mass were concentrated at its geometric centre. This allows us to treat planets, stars, and moons as point masses for calculating the forces between them, greatly simplifying cosmic mechanics.
The Universal Gravitational Constant (G)
The constant 'G' in the equation is the Universal Gravitational Constant. It is a fundamental constant of nature, meaning it is believed to be the same everywhere in the universe. Its extremely small value ($6.67 \times 10^{-11} \text{ N m}^2 \text{kg}^{-2}$) explains why gravitational forces are only noticeable when massive objects are involved. The value of G was first measured with reasonable accuracy by Henry Cavendish in 1798 using a sensitive torsion balance, an experiment often called 'weighing the Earth' because it allowed for the calculation of Earth's mass for the first time.
Worked examples
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Calculate the gravitational force between Earth (mass $5.97 \times 10^{24}$ kg) and the Moon (mass $7.35 \times 10^{22}$ kg) if their average centre-to-centre distance is $3.84 \times 10^8G = 6.67 \times 10^{-11} \text{ N m}^2 \text{kg}^{-2}
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Identify given values:
Two large spheres with masses kg and kg are placed with their centres 4.0 m apart. A smaller object with mass kg is placed on the line connecting their centres, at a distance of 1.0 m from the 200 kg sphere. Calculate the net gravitational force on the 10 kg object. Use .
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Identify forces and distances: The 10 kg mass (m) is attracted by both and . Let be the force from and be the force from . These forces act in opposite directions along the line connecting the centres.
How it all connects
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Glossary
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Revision flashcards
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What is the fundamental nature of gravitational force?
It is an attractive, non-contact force that exists between any two objects possessing mass.
Key takeaways
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13.2 Gravitational force between point masses.
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For a point outside a uniform sphere, the mass of the sphere may be considered to be a point mass at its centre.
- ✓
A uniform sphere is one where its mass is distributed evenly .
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The gravitational field lines around a uniform sphere are therefore identical to those around a point mass.
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An object can be regarded as point mass when a body covers a very large distance as compared to its size .
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Radial fields are considered non-uniform fields.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
9702/42 · Q1(a)(i)
For point X, determine: (i) the speed
9702/41 · Q5(c)(i)
Show that the distance y of point P from the centre of sphere Y is equal to 2x.
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Checkpoint
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