Basics of Gravitational Fields
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| Дата: | пʼятниця 25 вересня 2026 01:05 AM |
1. Introduction to Gravitation
Learning Outcomes
- I can describe gravity as an attractive force between masses.
- I can explain how gravity influences objects on Earth and in space.
- I can distinguish gravity from other fundamental forces.
- I can identify situations where gravitational forces are important.
- I can explain why gravity governs the motion of planets, moons, and satellites.
What Is Gravity?
Gravity is an attractive interaction between objects that have mass.
Every object with mass gravitationally attracts every other object with mass.
This means gravity exists between:
- you and Earth
- Earth and the Moon
- Earth and the Sun
- two people standing beside each other
- planets and their moons
- stars and planets
- galaxies and other galaxies
The gravitational attraction between ordinary objects is extremely small, so we usually notice gravity only when at least one of the objects has a very large mass.
Gravity Is Always Attractive
In ordinary physics, gravitational forces between masses are attractive.
This means two masses gravitationally pull toward one another.
For example, Earth attracts you toward its centre.
At the same time, you also gravitationally attract Earth.
The forces act in opposite directions along the line joining the centres of the two objects.
Because Earth's mass is enormous compared with a person's mass, Earth's resulting acceleration is immeasurably small in everyday situations.
Gravity Acts at a Distance
Gravity is a non-contact interaction.
Earth does not need to physically touch the Moon to affect its motion.
The Sun does not need to touch Earth to keep Earth in orbit.
Gravity acts across space.
This distinguishes gravity from contact forces such as:
- friction
- normal force
- tension from a rope
- air resistance
- spring forces
Gravitational Fields
A useful way to describe gravity is through a gravitational field.
A mass produces a gravitational field in the space around it.
Another mass placed within that field experiences a gravitational force.
For a spherical object such as Earth, the gravitational field points approximately toward the object's centre.
The field becomes weaker as distance from the object increases.
Gravity on Earth
Earth has a very large mass:
approximately 5.97 × 10²⁴ kg
Its gravitational field strongly influences objects near its surface.
Gravity causes objects to:
- fall when released
- have weight
- accelerate downward
- follow curved paths when thrown
- remain near Earth's surface
Near Earth's surface, the gravitational field strength is approximately:
g = 9.8 N/kg
For many introductory calculations, this may be rounded to:
g ≈ 10 N/kg
Mass and Weight
Mass and weight are related but different.
Mass measures the amount of matter or inertia of an object.
Unit:
kilogram (kg)
Weight is the gravitational force acting on an object.
Unit:
newton (N)
The relationship is:
W = mg
where:
- W = weight in newtons
- m = mass in kilograms
- g = gravitational field strength in N/kg
Example: Weight on Earth
A student has a mass of:
60 kg
Using:
g = 9.8 N/kg
Weight:
W = mg
W = 60 × 9.8
W = 588 N
The student's mass is:
60 kg
The student's weight is approximately:
588 N
Mass Does Not Depend on Location
Suppose an astronaut has a mass of:
70 kg
On Earth:
mass = 70 kg
On the Moon:
mass = 70 kg
In orbit:
mass = 70 kg
Mass does not change simply because the astronaut moves somewhere else.
Weight can change because gravitational field strength changes.
Weight on the Moon
The Moon has less mass than Earth, so its surface gravitational field is weaker.
The Moon's surface gravitational field strength is approximately:
1.6 N/kg
For a 60 kg person:
W = mg
W = 60 × 1.6
W = 96 N
The person's mass remains 60 kg, but their weight is much smaller.
Falling Objects
If an object is released near Earth's surface, gravity causes it to accelerate downward.
Ignoring air resistance, the acceleration is approximately:
9.8 m/s²
This is called the acceleration due to gravity.
The symbol:
g
is commonly used for both gravitational field strength and acceleration due to gravity.
Numerically near Earth's surface:
g ≈ 9.8 N/kg
and:
g ≈ 9.8 m/s²
These units describe closely related aspects of the same gravitational field.
Do Heavier Objects Fall Faster?
Ignoring air resistance, objects near Earth's surface experience the same gravitational acceleration regardless of their mass.
A heavy ball and a light ball released together in a vacuum fall with the same acceleration.
Why?
A more massive object experiences a greater gravitational force, but it also has greater inertia.
These effects balance so that the acceleration is the same when other forces are negligible.
Air Resistance Changes What We Observe
A feather and a metal ball do not usually fall together through Earth's atmosphere.
The reason is not that gravity stops working normally.
The difference comes from:
air resistance
The feather experiences air resistance that is large compared with its weight.
In a vacuum, where air resistance is absent, they fall together.
Gravity and Projectile Motion
Throw a ball horizontally.
Gravity immediately begins accelerating it downward.
The ball therefore follows a curved path.
Without gravity, the ball would continue moving approximately in a straight line at constant velocity if no other forces acted.
Gravity continually changes the vertical component of its velocity.
Gravity Extends Far Beyond Earth
Earth's gravitational field does not suddenly stop at the edge of the atmosphere.
It extends far into space.
As distance increases, the gravitational attraction becomes weaker, but it does not abruptly become zero.
This is why Earth's gravity can influence:
- the Moon
- artificial satellites
- spacecraft
- nearby objects in space
Newton's Universal Law of Gravitation
Newton described gravitational attraction mathematically.
For two masses:
F = Gm₁m₂/r²
where:
- F = gravitational force
- G = universal gravitational constant
- m₁ = first mass
- m₂ = second mass
- r = distance between their centres
This equation is called Newton's law of universal gravitation.
What Does the Gravitation Equation Tell Us?
The equation:
F = Gm₁m₂/r²
reveals two major relationships.
Gravitational force depends on:
1. The masses of the objects
Greater masses produce stronger gravitational attraction.
2. The distance between them
Greater distance produces weaker gravitational attraction.
Effect of Mass
Suppose one object's mass doubles while everything else stays constant.
Since:
F ∝ m
the gravitational force doubles.
If both masses double:
2 × 2 = 4
the gravitational force becomes:
4 times greater
Effect of Distance
Gravity follows an inverse-square relationship.
F ∝ 1/r²
If distance doubles:
F becomes 1/2² = 1/4
of its original value.
If distance triples:
F becomes 1/3² = 1/9
of its original value.
If distance becomes four times greater:
F becomes 1/16
of its original value.
Gravity weakens rapidly with increasing distance.
Why Don't We Notice Gravity Between People?
You and another person gravitationally attract each other.
However, human masses are relatively small.
The gravitational attraction between two people is therefore tiny compared with forces such as:
- Earth's gravity
- friction
- normal forces
- muscular forces
Gravity becomes especially important when astronomical masses are involved.
Gravity and the Moon
Earth gravitationally attracts the Moon.
So why does the Moon not simply fall directly into Earth?
Because the Moon also has a large sideways velocity.
Gravity continually bends the Moon's motion toward Earth.
Instead of travelling in a straight line, the Moon follows an orbit.
The Moon can therefore be thought of as continually falling around Earth.
Newton's Cannon Thought Experiment
Imagine firing a cannonball horizontally from a very high mountain.
At low speed, it travels forward and falls to Earth.
At greater speed, it travels farther before hitting Earth.
At a sufficiently high horizontal speed, Earth's surface curves away beneath it at the same rate that it falls.
The object enters orbit.
This thought experiment shows the connection between:
falling objects and orbiting objects
Both are influenced by gravity.
Gravity Provides the Centripetal Force
An orbiting object continuously changes direction.
A force directed toward the centre of its curved path is required.
For a satellite orbiting Earth, gravity provides this centripetal force.
For a simplified circular orbit:
Fgravity = Fcentripetal
Therefore:
GMm/r² = mv²/r
This relationship connects gravity with orbital motion.
Orbital Speed
Simplifying the circular-orbit equation gives:
v = √(GM/r)
This shows that orbital speed depends on:
- the mass of the central object
- the orbital distance
For satellites around the same planet, a satellite in a lower circular orbit generally moves faster than one in a higher circular orbit.
Artificial Satellites
Artificial satellites orbit Earth for many purposes.
Examples include:
- communications
- weather monitoring
- navigation
- Earth observation
- scientific research
Gravity continually accelerates these satellites toward Earth while their sideways motion carries them forward.
Astronauts in Orbit
Astronauts aboard an orbiting spacecraft often appear to be weightless.
This does not mean there is no gravity.
Earth's gravity is still acting strongly enough to maintain the spacecraft's orbit.
The astronauts and spacecraft are falling together.
This condition is called free fall.
Apparent Weightlessness
When standing on Earth, the ground pushes upward on you with a normal force.
You experience this support force as part of your apparent weight.
In an orbiting spacecraft:
- the spacecraft falls toward Earth
- the astronaut falls toward Earth
- both accelerate together
There is little or no normal force supporting the astronaut.
The astronaut therefore experiences apparent weightlessness.
Gravity and the Solar System
The Sun contains most of the mass in the Solar System.
Its gravitational influence governs the large-scale orbital motion of:
- planets
- dwarf planets
- asteroids
- comets
- many smaller objects
Planets have sideways velocities while the Sun's gravity continually changes their direction.
The result is orbital motion.
Planetary Orbits
Planetary orbits are not perfect circles.
They are approximately elliptical.
The Sun lies at one focus of a planet's elliptical orbit.
The planet's speed changes as its distance from the Sun changes.
This behaviour was described by Kepler's laws of planetary motion and later explained using Newtonian gravitation.
Moons and Planets
Planets can also gravitationally control the motion of nearby moons.
For example:
Earth's gravity governs the Moon's orbit.
Jupiter's gravity governs the orbits of its many moons.
Saturn's gravity governs its moons and strongly influences the particles forming its rings.
Gravity therefore creates systems within larger gravitational systems.
Gravity and Tides
The Moon's gravity affects Earth as well.
Differences in the Moon's gravitational pull across Earth contribute strongly to ocean tides.
The Sun also contributes to tides.
Tides provide an everyday example of gravitational effects between astronomical objects.
Gravity and Stars
Gravity is essential to stars.
A star contains an enormous amount of matter.
Gravity pulls the star's material inward.
Inside an active star, this inward gravitational effect is balanced by pressure associated with the hot gas and energy generated in the star.
Gravity therefore plays a major role in:
- star formation
- stellar structure
- stellar evolution
Gravity and Star Formation
Stars form from enormous clouds of gas and dust.
If part of a cloud becomes sufficiently dense, gravitational attraction can cause material to collapse inward.
As the material collapses:
- density increases
- pressure increases
- temperature increases
Eventually, conditions may become sufficient for nuclear fusion to begin.
Gravity therefore plays a fundamental role in creating stars.
Gravity on Larger Scales
Gravity also affects structures far larger than the Solar System.
It influences:
- star clusters
- galaxies
- galaxy groups
- galaxy clusters
Because gravity acts over enormous distances and is always attractive in ordinary matter, its effects accumulate on astronomical scales.
Gravity and the Four Fundamental Interactions
Modern physics describes four fundamental interactions:
- gravitational
- electromagnetic
- strong nuclear
- weak nuclear
Gravity is one of these fundamental interactions.
The others behave very differently.
Gravity vs Electromagnetism
Gravity:
- acts between masses/energy
- is attractive between ordinary masses
- has effectively unlimited range
- is comparatively very weak at the particle scale
- dominates many astronomical systems
Electromagnetism:
- acts between electrically charged particles
- can attract or repel
- has unlimited range
- is much stronger than gravity at atomic scales
- governs electricity, magnetism, chemistry, and many contact forces
Gravity vs the Strong Interaction
The strong interaction acts on subatomic particles and is responsible for binding quarks and, through the residual strong interaction, helping bind protons and neutrons in atomic nuclei.
It is:
- extremely strong at nuclear scales
- effective only over very short distances
Gravity is much weaker at the particle level but acts across enormous distances.
Gravity vs the Weak Interaction
The weak interaction is involved in processes such as certain forms of radioactive decay.
It operates over extremely short distances.
Gravity, by contrast, has long-range effects and becomes dominant when dealing with very large astronomical masses.
Why Can the Weakest Force Dominate Space?
At the particle scale, gravity is extremely weak compared with electromagnetism and the nuclear interactions.
So why does gravity dominate planets and stars?
One major reason is that large astronomical objects are generally close to electrically neutral overall.
Positive and negative electric charges tend to cancel.
Mass does not cancel in the same way.
As enormous amounts of matter accumulate, their gravitational effects also accumulate.
Gravity Is Not the Same as Magnetism
Gravity and magnetism are sometimes confused because both can act without direct contact.
However, they are different interactions.
Gravity:
acts between masses
Magnetic forces:
are associated with magnetic fields, moving charges, and magnetic materials
A rock falls toward Earth because of gravity, not because Earth is acting like a magnet on the rock.
Gravity Is Not the Normal Force
A book resting on a table experiences gravity downward.
But it does not accelerate downward because the table provides an upward normal force.
If the book is stationary:
Fnet = 0
and approximately:
N = W
The normal force does not replace gravity. Both forces act simultaneously.
Gravity Is Not Air Resistance
A falling object may experience:
weight downward
and:
air resistance upward
Gravity and air resistance are different forces.
Gravity exists even in a vacuum.
Air resistance requires interaction with particles in a gas.
Gravitational Force Between Two Objects
Consider two objects:
m₁ = 5 kg
m₂ = 10 kg
separated by some distance.
Each object gravitationally attracts the other.
According to Newton's third law:
the force on object 1 due to object 2 is equal in magnitude and opposite in direction to the force on object 2 due to object 1.
Even if the masses are very different, the interaction forces form an equal-and-opposite pair.
Worked Example 1: Weight
A 50 kg student stands on Earth.
Using:
g = 9.8 N/kg
Calculate weight:
W = mg
W = 50 × 9.8
W = 490 N
Worked Example 2: Different Planet
A 40 kg object is placed where:
g = 3.7 N/kg
Calculate its weight:
W = 40 × 3.7
W = 148 N
Its mass remains:
40 kg
Worked Example 3: Doubling Mass
Two objects experience gravitational force F.
If one object's mass doubles while everything else remains constant:
Fnew = 2F
The gravitational force doubles.
Worked Example 4: Doubling Both Masses
If both masses double:
Fnew = (2)(2)F
Therefore:
Fnew = 4F
The gravitational force becomes four times larger.
Worked Example 5: Doubling Distance
Two masses experience force F at distance r.
If the distance becomes:
2r
then:
Fnew = F/2²
Therefore:
Fnew = F/4
The gravitational force becomes one quarter as large.
Worked Example 6: Tripling Distance
If the distance becomes:
3r
then:
Fnew = F/3²
Therefore:
Fnew = F/9
The gravitational force becomes one ninth as large.
Worked Example 7: Combining Changes
Suppose both masses double and their separation also doubles.
Mass effect:
2 × 2 = 4
Distance effect:
1/2² = 1/4
Combine:
4 × 1/4 = 1
Therefore:
Fnew = F
The gravitational force remains unchanged.
Worked Example 8: Orbital Motion
A satellite travels sideways around Earth.
Without gravity, it would tend to continue along a straight-line path.
Earth's gravity continuously changes the satellite's direction toward Earth.
The combination produces:
orbital motion
The satellite is continually falling toward Earth while moving forward.
Worked Example 9: Apparent Weightlessness
An astronaut and spacecraft are orbiting Earth.
Both experience Earth's gravitational field.
Both accelerate toward Earth together.
Because the astronaut is not being supported by a surface in the usual way, the astronaut experiences:
apparent weightlessness
This does not mean:
gravity = 0
Worked Example 10: Comparing Forces
A book sits on a desk.
Forces on the book include:
gravity downward
and:
normal force upward
These forces may be equal in magnitude when the book is stationary.
However, they are different types of forces:
- gravity is a long-range fundamental interaction
- the normal force is a contact force arising from electromagnetic interactions between matter
Gravity in Everyday Situations
Gravity is important when:
- an object falls
- a ball is thrown
- a person jumps
- rain falls
- water flows downhill
- an object has weight
- a pendulum swings
- ocean tides occur
Although gravity is always present, other forces may balance or oppose it.
Gravity in Space
Gravity is important when:
- planets orbit stars
- moons orbit planets
- satellites orbit Earth
- comets travel through the Solar System
- stars form
- galaxies interact
- spacecraft change trajectories
There is no sharp boundary beyond which gravity simply disappears.
Common Mistakes
Mistake 1: "There is no gravity in space."
Incorrect.
Gravity acts throughout space. Astronauts in orbit appear weightless because they are in continuous free fall.
Mistake 2: "Heavy objects always fall faster."
Ignoring air resistance, objects experience the same gravitational acceleration near Earth's surface.
Mistake 3: "Mass and weight are the same."
Mass is measured in:
kg
Weight is a force measured in:
N
Mistake 4: "Gravity only acts downward."
"Downward" near a planet means:
toward the planet's centre
In space, gravitational force points toward the attracting mass.
Mistake 5: "Satellites are beyond Earth's gravity."
Satellites orbit precisely because Earth's gravity still acts on them.
Mistake 6: "Orbiting objects have no acceleration."
An orbiting object's direction is continuously changing.
A change in velocity means there is acceleration.
Error Analysis
A student says:
"The Moon is not falling because it stays the same distance from Earth."
This is misleading.
The Moon is continually accelerated toward Earth by gravity.
However, it also has sideways velocity.
The combination produces its orbit.
The Moon can therefore be understood as continually falling around Earth rather than falling directly into it.
Another Error Analysis
A student says:
"An astronaut has no weight in orbit because Earth has no gravitational field there."
This is incorrect.
Earth's gravitational field extends into space.
The astronaut appears weightless because the astronaut and spacecraft are falling together.
The correct concept is:
free fall and apparent weightlessness
A Useful Gravity Problem-Solving Strategy
When analysing a gravitational situation:
Step 1: Identify the objects interacting.
Step 2: Identify which mass is producing the important gravitational field.
Step 3: Determine the direction of gravitational force.
Step 4: Decide whether other forces are present.
Step 5: Choose the appropriate relationship, such as W = mg or Newton's gravitation equation.
Step 6: Consider how mass and distance affect the force.
Step 7: Check whether the result makes physical sense.
Did You Know?
The same basic gravitational interaction helps explain phenomena ranging from a falling apple to the motion of moons, planets, stars, and galaxies.
Newton's great insight was that the gravity causing objects to fall near Earth's surface and the gravity governing the Moon's orbit could be understood as manifestations of the same universal interaction.
This connected terrestrial physics and astronomical motion within one mathematical framework.
Key Terms
- Gravity: Attractive interaction associated with mass and energy.
- Gravitation: Gravitational interaction between masses.
- Gravitational force: Attractive force between masses in Newtonian physics.
- Gravitational field: Region in which a mass experiences gravitational influence.
- Gravitational field strength: Gravitational force per unit mass, measured in N/kg.
- Mass: Measure of an object's inertia, measured in kilograms.
- Weight: Gravitational force acting on an object.
- Free fall: Motion when gravity is the only significant force acting.
- Orbit: Curved path of one object around another due to gravity and its motion.
- Satellite: Object orbiting another object.
- Natural satellite: Naturally occurring orbiting object, such as the Moon.
- Artificial satellite: Human-made object placed into orbit.
- Centripetal force: Net inward force required for circular motion.
- Inverse-square relationship: Relationship in which a quantity varies as 1/r².
- Apparent weightlessness: Condition experienced when an object and its surroundings are in free fall together.
- Fundamental interaction: One of the basic interactions used to describe physical phenomena.
Key Equations
Weight:
W = mg
Newton's law of universal gravitation:
F = Gm₁m₂/r²
For circular orbital motion:
Fgravity = Fcentripetal
and:
GMm/r² = mv²/r
Circular orbital speed:
v = √(GM/r)
Near Earth's surface:
g ≈ 9.8 N/kg
and:
g ≈ 9.8 m/s²
Key Relationships
If one mass doubles:
F → 2F
If both masses double:
F → 4F
If distance doubles:
F → F/4
If distance triples:
F → F/9
If distance becomes four times greater:
F → F/16
Gravity therefore:
increases with mass
and:
decreases with the square of distance
Key Takeaways
- Gravity is an attractive interaction between objects with mass.
- Every mass gravitationally attracts every other mass.
- Gravity is a non-contact interaction.
- A mass creates a gravitational field in the space around it.
- Near a spherical planet, the gravitational field points approximately toward the planet's centre.
- Gravity affects objects both on Earth and throughout space.
- Earth's gravity gives objects weight.
- Mass and weight are different quantities.
- Mass is measured in kilograms.
- Weight is measured in newtons.
- Weight can be calculated using W = mg.
- Mass normally remains constant when an object moves to another location, while weight can change.
- Near Earth's surface, gravitational field strength is approximately 9.8 N/kg.
- Ignoring air resistance, objects near Earth fall with the same gravitational acceleration.
- Gravity follows an inverse-square relationship with distance in Newtonian physics.
- Increasing mass increases gravitational attraction.
- Increasing separation decreases gravitational attraction.
- Gravity does not suddenly disappear in space.
- Astronauts in orbit experience apparent weightlessness because they and their spacecraft are in continuous free fall.
- The Moon remains in orbit because gravity continually changes its direction while it has sideways velocity.
- Artificial satellites orbit Earth for the same fundamental reason.
- Gravity provides the centripetal force required for many orbital motions.
- The Sun's gravity governs the large-scale orbital motion of planets and many smaller Solar System objects.
- Planetary orbits are approximately elliptical rather than perfectly circular.
- Gravity also influences tides, star formation, galaxies, and galaxy clusters.
- Gravity is one of the four fundamental interactions.
- Gravity differs from electromagnetic, strong, and weak interactions in its properties and relative strength.
- Contact forces such as the normal force and friction are not forms of gravity.
- Gravity is comparatively weak at particle scales but becomes dominant across many astronomical systems.
- The same gravitational principles connect falling objects on Earth with the motion of planets, moons, and satellites.
2. Newton's Law of Gravitation
Learning Outcome
- I can state Newton's Law of Universal Gravitation.
- I can identify the variables that affect gravitational force.
- I can explain how mass influences gravitational attraction.
- I can calculate gravitational force between two objects.
- I can apply Newton's Law of Gravitation to real-world situations.
- Mathematical formulation: \( F = G \frac{m_1m_2}{r^2} \)
- Units and dimensions of G
Newton's Great Discovery
Before the 17th century, people knew that objects fell to Earth, but they did not understand why planets stayed in orbit around the Sun.
In 1687, Sir Isaac Newton proposed that the same force responsible for an apple falling from a tree also keeps the Moon in orbit around Earth and the planets in orbit around the Sun.
He called this force gravitation and described it with one of the most important equations in physics—Newton's Law of Universal Gravitation.
The word universal means that the law applies everywhere in the universe, from tiny objects on Earth to galaxies billions of light-years away.
Newton's Law of Universal Gravitation
Newton's Law states:
Every object in the universe attracts every other object with a gravitational force.
The strength of this force depends on:
- The masses of the two objects.
- The distance between their centres.
The mathematical relationship is:
where:
- F = gravitational force (N)
- G = universal gravitational constant
- m₁ = mass of the first object (kg)
- m₂ = mass of the second object (kg)
- r = distance between the centres of the two objects (m)
The gravitational constant is:
This very small value explains why gravitational forces between everyday objects are usually too weak to notice.
Variables that Affect Gravitational Force
There are only two factors that determine the strength of gravitational attraction.
1. Mass
Larger masses produce stronger gravitational forces.
If either mass increases:
- The gravitational force increases.
If one mass doubles:
- The force doubles.
If both masses double:
- The force becomes four times larger.
2. Distance
Gravity becomes weaker as objects move farther apart.
However, gravity does not decrease in a simple linear way.
Instead, it follows the inverse-square law.
This means:
If the distance doubles:
- The force becomes one-quarter as large.
If the distance triples:
- The force becomes one-ninth as large.
If the distance becomes four times larger:
- The force becomes one-sixteenth as large.
| Distance Change | New Gravitational Force |
|---|---|
| Double distance | 1/4 of original |
| Triple distance | 1/9 of original |
| Four times farther | 1/16 of original |
| Half the distance | 4 times greater |
Notice that distance has a much greater effect than mass.
The Universal Gravitational Constant
The constant G is the same everywhere in the universe.
Its small value shows that gravity is an extremely weak force between small objects.
For example:
Two students standing 1 metre apart attract one another gravitationally—but the force is so tiny that it is completely overwhelmed by friction and other everyday forces.
Gravity only becomes dominant when enormous masses such as planets and stars are involved.
Equal and Opposite Forces
Newton's Law of Gravitation follows Newton's Third Law of Motion.
If Earth pulls on the Moon with a certain force:
- The Moon pulls on Earth with exactly the same force.
The forces are equal in magnitude but act on different objects.
The reason Earth moves much less is that it has far greater mass.
Worked Example
Question
Calculate the gravitational force between two objects.
Object A:
Mass = 10 kg
Object B:
Mass = 20 kg
Distance between centres = 2.0 m
Step 1
Write the equation.
Step 2
Substitute the values.
Step 3
Calculate.
Answer:
The gravitational force is approximately:
3.3×10−9 N
This extremely small force illustrates why gravity between everyday objects is difficult to detect.
Real-World Applications
Newton's Law of Gravitation is used to predict and understand many important phenomena.
Examples include:
Planetary Motion
Calculating the forces keeping planets in orbit around the Sun.
Moon Orbits
Predicting the Moon's motion around Earth.
Artificial Satellites
Designing satellite orbits for:
- GPS
- Weather forecasting
- Communications
- Earth observation
Space Missions
Planning spacecraft trajectories to the Moon, Mars, and beyond.
Binary Stars
Calculating how pairs of stars orbit one another.
Galaxies
Studying how galaxies rotate and interact through gravity.
Why Gravity Controls the Solar System
The Sun contains about 99.8% of the Solar System's total mass.
Because gravitational force increases with mass:
- The Sun exerts an enormous gravitational pull on every planet.
Each planet moves forward while simultaneously being pulled toward the Sun.
This continuous inward pull causes the planets to travel in stable orbits rather than moving away into space.
Limitations of Newton's Theory
Newton's Law accurately predicts the motion of most objects in the Solar System.
However, for:
- Extremely strong gravitational fields,
- Objects moving near the speed of light,
Albert Einstein's General Theory of Relativity provides more accurate predictions.
Even so, Newton's Law remains an excellent approximation for almost all engineering and everyday astronomical calculations.
Key Terms
Universal Gravitation — The attractive force between any two objects with mass.
Gravitational Constant (G) — A constant that determines the strength of gravitational attraction.
Inverse-Square Law — A relationship in which a quantity decreases in proportion to the square of the distance.
Centre of Mass — The point at which an object's mass is considered to be concentrated for gravitational calculations.
Key Takeaways
- Every pair of objects with mass attracts one another.
- Gravitational force depends on mass and distance.
- Larger masses produce stronger gravitational forces.
- Gravitational force decreases with the square of the distance between objects.
- Gravity between everyday objects is extremely small because the gravitational constant is very small.
- Newton's Law of Gravitation explains the motion of planets, moons, satellites, and many other astronomical systems.
Suggested Images
Suggested placement:
- After "Newton's Great Discovery" – Illustration of Isaac Newton and the famous falling apple, introducing the historical context.
- After "Newton's Law of Universal Gravitation" – Diagram of two masses with attractive force arrows acting along the line joining their centres.
- After "Distance" – Inverse-square law illustration comparing the gravitational force at different separations.
- After "Why Gravity Controls the Solar System" – Solar System diagram emphasizing the Sun's dominant gravitational influence on the planets.
3. Gravitational Constant
Learning Outcomes
- I can define the gravitational constant, .
- I can explain the role of in Newton's Law of Gravitation.
- I can identify the units of the gravitational constant.
- I can use in gravitational force calculations.
- I can explain why the value of makes gravity relatively weak compared to other forces.
- Cavendish experiment
- Values and implications of G
What is the Gravitational Constant?
The gravitational constant, represented by the symbol G, is a fundamental constant of nature that determines the strength of the gravitational force between two masses.
It appears in Newton's Law of Universal Gravitation, allowing scientists to calculate the gravitational attraction between any two objects.
The value of the gravitational constant is:
G=6.67×10−11 N\cdotpm2/kg2This value is the same everywhere in the universe.
Whether calculating the attraction between two apples, Earth and the Moon, or distant stars, the same value of G is always used.
Why is G Called a Constant?
A constant is a quantity whose value never changes.
Unlike mass or distance, which vary from one situation to another, the gravitational constant has the same value:
- On Earth
- On the Moon
- Throughout the Solar System
- In distant galaxies
Because G is universal, Newton's Law can be applied anywhere in the universe.
The Role of G in Newton's Law
Newton's Law of Universal Gravitation is written as:
where:
- F = gravitational force (N)
- G = gravitational constant
- m₁ = first mass (kg)
- m₂ = second mass (kg)
- r = distance between the centres of the masses (m)
The gravitational constant acts as the proportionality constant that links the masses and their separation to the resulting gravitational force.
Without G, the equation would not produce the correct value for the force.
Units of the Gravitational Constant
The SI units of G are:
N\cdotpm2/kg2These units ensure that when:
- Mass is measured in kilograms (kg),
- Distance is measured in metres (m),
the calculated force is expressed in newtons (N).
Remember:
- Mass → kilograms (kg)
- Distance → metres (m)
- Force → newtons (N)
Using consistent SI units is essential for obtaining the correct answer.
Why is the Value of G So Small?
The value of G is extremely small:
This tiny number means that gravity is an inherently weak force between small objects.
For example:
Two students standing one metre apart attract each other gravitationally, but the force is far too small to be felt.
Other forces, such as friction and electromagnetic forces, completely dominate everyday situations.
Gravity is Weak—but Important
Although gravity is the weakest of the four fundamental forces, it becomes the dominant force on astronomical scales.
Why?
Because gravity has two unique properties:
- It acts between all objects with mass.
- It is always attractive.
Unlike electric forces, which can both attract and repel, gravitational forces always add together.
When enormous masses such as planets and stars are involved, gravity becomes incredibly strong.
For example:
- Earth's gravity holds the atmosphere in place.
- The Sun's gravity keeps the planets in orbit.
- Gravity binds galaxies together.
Using G in Calculations
Whenever Newton's Law of Gravitation is used, G must be included.
Worked Example
Question
Calculate the gravitational force between two masses.
Mass 1 = 500 kg
Mass 2 = 800 kg
Distance between centres = 10 m
Step 1
Write the equation.
Step 2
Substitute the values.
Step 3
Calculate.
First calculate the masses:
Then divide by:
Finally:
Answer
F=2.67×10−7 NAlthough these masses are quite large, the force remains extremely small because the value of G is so tiny.
Measuring G
The gravitational constant was first measured experimentally by the English scientist Henry Cavendish in 1798.
He used a very sensitive device called a torsion balance.
The experiment measured the tiny gravitational attraction between large lead spheres.
Cavendish's work allowed scientists to:
- Calculate Earth's mass.
- Better understand planetary motion.
- Improve gravitational calculations throughout astronomy.
His experiment is often described as "weighing the Earth."
Comparing Gravity with Other Forces
Gravity is much weaker than the other three fundamental forces.
For example:
- A small refrigerator magnet can easily lift a paperclip against the pull of the entire Earth.
This happens because the magnetic force between the magnet and the paperclip is much stronger than the gravitational force acting on the paperclip.
Although gravity dominates the motion of planets and stars, it is usually insignificant when dealing with atoms and molecules.
Why Scientists Still Study G
Although G has been known for more than 200 years, measuring it precisely remains surprisingly difficult.
Gravity between laboratory-sized objects is extremely weak.
Modern scientists continue improving measurements of G because accurate values are important for:
- Astronomy
- Satellite navigation
- Planetary science
- Cosmology
- Tests of fundamental physics
Key Terms
Gravitational Constant (G) — The universal constant that determines the strength of gravitational attraction between masses.
Universal Constant — A physical constant that has the same value everywhere in the universe.
Proportionality Constant — A constant that relates quantities in a mathematical equation.
Torsion Balance — A sensitive instrument used by Henry Cavendish to measure the gravitational constant.
Key Takeaways
- G is the universal gravitational constant.
-
Its value is:
- G appears in Newton's Law of Universal Gravitation.
- The very small value of G explains why gravity is much weaker than the other fundamental forces.
- Gravity only becomes dominant when extremely large masses, such as planets and stars, are involved.
- Scientists use G to calculate gravitational forces throughout the universe.
Suggested Images
Suggested placement:
- After "Measuring G" – Illustration or historical drawing of Henry Cavendish's torsion balance experiment.
- After "The Role of G in Newton's Law" – Diagram of two masses separated by a distance with force arrows and the gravitation equation.
- After "Comparing Gravity with Other Forces" – Photo of a magnet lifting a paperclip to illustrate how electromagnetic forces can easily overcome gravity.
- After "Gravity is Weak—but Important" – Earth–Moon system showing gravitational attraction acting between two massive bodies.
4. Inverse-Square Law
Learning Outcomes
- I can explain the inverse-square relationship in gravitation.
- I can predict how gravitational force changes with distance.
- I can calculate force changes when distance is doubled or halved.
- I can interpret graphs showing inverse-square relationships.
- I can apply the inverse-square law to planetary and satellite systems.
- Inverse-square law derivation
- Gravitational field examples
What is the Inverse-Square Law?
One of the most important features of gravity is that its strength decreases rapidly as the distance between two objects increases.
This relationship is called the inverse-square law.
The inverse-square law states:
The gravitational force between two objects is inversely proportional to the square of the distance between their centres.
This means that even a small increase in distance can produce a large decrease in gravitational force.
The Mathematics of the Inverse-Square Law
Newton's Law of Universal Gravitation is:
Notice that the distance r appears in the denominator as r².
This means:
The force does not decrease in direct proportion to distance—it decreases with the square of the distance.
How Distance Affects Gravity
Suppose the distance between two objects changes while their masses remain constant.
If the distance doubles
The force becomes:
The gravitational force becomes one-quarter of its original value.
If the distance triples
The force becomes:
91Only one-ninth of the original force remains.
If the distance becomes four times larger
The force becomes:
161Only one-sixteenth of the original force remains.
If the distance is halved
The force becomes:
4FThe gravitational force becomes four times stronger.
Summary Table
| Change in Distance | Change in Gravitational Force |
|---|---|
| Double (×2) | 1/4 as large |
| Triple (×3) | 1/9 as large |
| Four times farther (×4) | 1/16 as large |
| Half (×1/2) | 4 times larger |
| One-third (×1/3) | 9 times larger |
This demonstrates how strongly gravity depends on distance.
Why Does Gravity Decrease So Quickly?
Imagine a light bulb shining equally in all directions.
As you move farther away:
- The same amount of light spreads over a larger area.
- The light appears dimmer.
Gravity behaves in a similar way.
As the distance increases, the gravitational influence spreads over a larger spherical area.
Since the surface area of a sphere increases as:
the strength of gravity decreases with the square of the distance.
Worked Example 1
Two planets exert a gravitational force of 800 N on each other.
If the distance between them doubles, what is the new force?
Step 1
Doubling the distance gives:
41of the original force.
Step 2
Calculate:
Answer
200 NWorked Example 2
A satellite experiences a gravitational force of 120 N.
It moves to half its original distance from Earth.
What is the new force?
Step 1
Halving the distance increases the force by:
4Step 2
Calculate:
Answer
480 NInterpreting Graphs
A graph of gravitational force against distance has a distinctive shape.
It is not a straight line.
Instead:
- The force decreases rapidly at small distances.
- The curve gradually flattens as distance increases.
- The graph approaches zero but never actually reaches it.
This type of curve is called an inverse-square curve.
Unlike a linear graph, equal increases in distance do not produce equal decreases in force.
Applications in the Solar System
The inverse-square law explains why:
Planets farther from the Sun experience weaker gravity.
For example:
- Mercury experiences a much stronger gravitational pull from the Sun than Neptune.
Satellites
As a satellite moves farther from Earth:
- Earth's gravitational pull decreases.
- The satellite moves more slowly in its orbit.
This is why satellites in higher orbits take longer to complete one revolution.
Moons
The same law explains why moons orbiting far from their planets experience weaker gravitational forces than nearby moons.
Why the Inverse-Square Law Matters
Scientists use the inverse-square law to:
- Predict planetary orbits.
- Calculate satellite trajectories.
- Design space missions.
- Study binary star systems.
- Understand galaxy interactions.
Without the inverse-square law, modern astronomy and space exploration would not be possible.
Common Mistakes
Students often confuse an inverse relationship with an inverse-square relationship.
Incorrect thinking
"If the distance doubles, the force halves."
❌ This is wrong.
Correct thinking
"If the distance doubles, the force becomes one-quarter."
✔ Because the distance is squared.
Always remember:
Square the distance first.
Key Terms
Inverse-Square Law — A relationship in which a quantity decreases in proportion to the square of the distance.
Inverse Proportion — A relationship in which one quantity decreases as another increases.
Relative Force — The force expressed as a fraction or multiple of another force.
Key Takeaways
- Gravitational force follows the inverse-square law.
- Doubling the distance reduces the force to one-quarter.
- Halving the distance increases the force by four times.
- The inverse-square law produces a curved graph rather than a straight line.
- This relationship explains the motion of planets, moons, and satellites.
- The inverse-square law is fundamental to astronomy, astrophysics, and space exploration.
Suggested Images
Suggested placement:
- After "How Distance Affects Gravity" – Diagram comparing two masses at distances r, 2r, and 3r, with arrows showing the decreasing gravitational force.
- After "Why Does Gravity Decrease So Quickly?" – Illustration of expanding spheres (or light spreading from a bulb) to visualize why intensity and gravitational influence decrease withr2.
- After "Applications in the Solar System" – Diagram of the Solar System highlighting that planets farther from the Sun experience weaker gravitational attraction.
- After "Satellites" – Comparison of a low-Earth orbit satellite and a high-altitude satellite, illustrating weaker gravity and longer orbital periods at greater distances.
5. Applications of Gravitation
Learning Outcomes
- I can identify real-world applications of gravitational theory.
- I can explain how gravity affects planetary motion.
- I can describe the role of gravity in satellite systems.
- I can analyze examples involving tides and celestial mechanics.
- I can connect gravitational concepts to astronomy and space exploration.
- Celestial mechanics
- Tides and gravity
Why is Gravitation Important?
Gravity is one of the most important forces in the universe.
Although it is the weakest of the four fundamental forces, gravity dominates the motion of planets, moons, stars, galaxies, and spacecraft because it acts over enormous distances and always attracts.
Our understanding of gravity allows scientists and engineers to:
- Predict planetary motion.
- Launch and operate satellites.
- Explore the Solar System.
- Study distant galaxies.
- Understand tides.
- Search for planets around other stars.
Modern astronomy and space exploration depend on the principles of gravitation.
Gravity and Planetary Motion
The planets remain in orbit around the Sun because of gravity.
The Sun contains about 99.8% of the mass of the Solar System, giving it an enormous gravitational pull.
As each planet moves forward through space, the Sun's gravity continuously pulls it inward.
The combination of:
- Forward motion (inertia)
- Gravitational attraction
causes the planets to travel in stable, curved paths called orbits.
Without gravity, the planets would continue moving in straight lines and drift away from the Solar System.
Kepler's Laws and Gravity
Johannes Kepler discovered that planets move in predictable ways around the Sun.
Later, Isaac Newton showed that gravity provides the force responsible for these motions.
Together, Kepler's observations and Newton's theory explain why:
- Planets follow elliptical orbits.
- Inner planets orbit more quickly than outer planets.
- Orbital periods increase with distance from the Sun.
Gravity provides the centripetal force that continually changes a planet's direction of motion.
Gravity and Satellites
Artificial satellites remain in orbit because gravity constantly pulls them toward Earth.
However, satellites also have a large sideways velocity.
Rather than falling directly to Earth, they continuously "fall around" the planet.
This creates a stable orbit.
Different satellite orbits serve different purposes.
Low Earth Orbit (LEO)
Typical altitude:
- 160–2,000 km
Used for:
- The International Space Station (ISS)
- Earth observation
- Scientific research
Medium Earth Orbit (MEO)
Used for:
- GPS navigation satellites
Geostationary Orbit (GEO)
Altitude:
Approximately 35,786 km
A satellite in geostationary orbit circles Earth once every 24 hours, matching Earth's rotation.
From the ground, it appears to remain fixed above one location.
These satellites are commonly used for:
- Television broadcasting
- Weather monitoring
- Communications
Gravity and Ocean Tides
The tides are caused primarily by the gravitational attraction of the Moon, with a smaller contribution from the Sun.
The Moon pulls more strongly on the side of Earth closest to it than on the far side.
This difference in gravitational pull creates two tidal bulges:
- A high tide on the side facing the Moon.
- A high tide on the opposite side.
As Earth rotates, most coastal areas experience approximately two high tides and two low tides each day.
Celestial Mechanics
Celestial mechanics is the study of the motion of objects in space under the influence of gravity.
Scientists use celestial mechanics to predict:
- Planetary orbits.
- Moon orbits.
- Asteroid paths.
- Comet trajectories.
- Spacecraft motion.
These calculations allow space agencies to accurately navigate missions across the Solar System.
Gravity Assists
One of the most remarkable applications of gravitation is the gravity assist, also called a gravitational slingshot.
A spacecraft flies close to a planet and uses the planet's gravity to change its speed and direction.
This allows spacecraft to:
- Reach distant planets more quickly.
- Save large amounts of fuel.
- Extend mission lifetimes.
Famous missions using gravity assists include:
- Voyager 1
- Voyager 2
- Cassini
- New Horizons
Gravity and Space Exploration
Every space mission depends on gravity.
Engineers use gravitational calculations to:
- Launch rockets into orbit.
- Place satellites at precise altitudes.
- Send spacecraft to other planets.
- Return astronauts safely to Earth.
- Predict planetary encounters years in advance.
Without accurate gravitational models, modern space exploration would not be possible.
Gravity in Astronomy
Astronomers cannot usually "see" gravity directly.
Instead, they observe its effects.
Gravity allows scientists to:
Measure Planetary Masses
By studying the motion of moons and satellites.
Discover Exoplanets
A planet causes its parent star to wobble slightly.
This tiny motion reveals the planet's presence.
Study Binary Stars
Pairs of stars orbit one another because of gravity.
Their orbital motion allows astronomers to calculate their masses.
Detect Black Holes
Although black holes emit almost no light, astronomers observe nearby stars orbiting an invisible object.
The stars' motions reveal the presence of an extremely massive black hole.
Gravity Shapes the Universe
Gravity is responsible for the formation and evolution of many astronomical structures.
It helps form:
- Stars
- Planets
- Solar systems
- Galaxies
- Galaxy clusters
Gravity also influences:
- The collisions of galaxies.
- The formation of black holes.
- The birth of new stars from giant clouds of gas and dust.
Without gravity, matter would not gather together to form the structures we observe in the universe today.
Everyday Applications
Although we often associate gravity with astronomy, it also affects everyday life.
Examples include:
- Keeping our feet on the ground.
- Holding Earth's atmosphere in place.
- Causing rivers to flow downhill.
- Producing waterfalls.
- Helping hydroelectric power stations generate electricity.
- Influencing the flight paths of rockets and aircraft.
Key Terms
Orbit — The curved path followed by an object under the influence of gravity.
Celestial Mechanics — The study of the motion of astronomical objects under gravitational forces.
Gravity Assist (Gravitational Slingshot) — A technique that uses the gravity of a planet to change the speed and direction of a spacecraft.
Geostationary Orbit — An orbit in which a satellite remains above the same point on Earth's surface by matching Earth's rotational period.
Tides — The regular rise and fall of sea levels caused mainly by the gravitational pull of the Moon and, to a lesser extent, the Sun.
Key Takeaways
- Gravity keeps planets, moons, and satellites in orbit.
- The Sun's gravity governs the motion of the Solar System.
- Artificial satellites rely on gravity to remain in stable orbits.
- The Moon's gravity is the primary cause of Earth's tides.
- Gravity assists allow spacecraft to travel farther while using less fuel.
- Astronomers use gravity to discover planets, measure stellar masses, detect black holes, and understand the evolution of the universe.
- Gravitation is fundamental to astronomy, satellite technology, and space exploration.
Suggested Images
Suggested placement:
- After "Gravity and Planetary Motion" – Solar System diagram illustrating planetary orbits around the Sun.
- After "Gravity and Satellites" – Comparison of Low Earth Orbit (LEO), Medium Earth Orbit (MEO), and Geostationary Orbit (GEO).
- After "Gravity and Ocean Tides" – Diagram showing the Moon's gravitational pull creating tidal bulges on Earth.
- After "Gravity Assists" – Spacecraft trajectory using a gravitational slingshot around Jupiter or another planet.
- Near "Gravity in Astronomy" – Illustrations of binary stars, exoplanet detection by stellar wobble, and stars orbiting a supermassive black hole.
- Near "Gravity and Space Exploration" – Rocket launch or spacecraft entering orbit to connect gravitational theory with modern space missions.