Gravitational Force

Lesson ~10 min read 8 MCQs

In simple terms: In simple terms, gravitational force is the universal attraction between any two objects with mass, from apples and planets to you and your pencil.

Why this matters

You've probably heard the story of Isaac Newton and the falling apple. It’s a great story, but it leaves out the most brilliant part of his discovery. Newton didn't just wonder why the apple fell down. He wondered why the Moon didn't fall down.

Think about it. The same force pulling the apple to the ground is also acting on the Moon. So why doesn't the Moon crash into Illinois?

The answer is that it is falling. It's constantly falling around the Earth. Newton's incredible insight was that the force of gravity isn't just a local thing here on Earth. It's universal. It's the invisible string that holds planets in orbit and galaxies together. In this lesson, we'll unpack this powerful idea, from the scale of the cosmos all the way down to why you feel lighter in an elevator.

Diagram

This diagram shows Newton's Law of Universal Gravitation. Two spherical masses, a larger m1 and a smaller m2, are shown in space. A dashed line labeled 'r' indicates the distance between their centers, and two equal and opposite force vectors labeled 'F' show the attractive gravitational force pulling each mass toward the other.

Concept map

flowchart TD
    A[Any two objects with mass m1, m2] --> B{Newton's Law of Universal Gravitation};
    B --> C["F_g = G(m1 * m2) / r^2"];
    C --> D{Is one mass a huge planet, M?};
    D -- Yes --> E[Creates a Gravitational Field];
    E --> F["g = GM / r^2"];
    F --> G{Is a small object 'm' in the field?};
    G -- Yes --> H[Object has Weight];
    H --> I["F_g = mg"];
    I --> J{Is the object accelerating vertically?};
    J -- Yes --> K[Apparent Weight != True Weight];
    K --> L["F_normal = mg + ma"];
    D -- No --> M[Force is usually negligible];

Core explanation

The Universe's Great Attraction

Every object with mass in the universe pulls on every other object with mass. You are gravitationally attracted to your desk, the person next to you, and the star system Alpha Centauri. So why don't you all just clump together? Because the force is incredibly weak unless at least one of the objects is massive—like, planet-massive.

Isaac Newton put this relationship into a beautiful, powerful equation called the Law of Universal Gravitation:

|F⃗_g| = G * (m₁ * m₂) / r²

Let's break that down:

F_g is the magnitude of the gravitational force between two objects.
m₁ and m₂ are the masses of the two objects. Notice they are multiplied—if you double one mass, you double the force.
r is the distance between the centers of mass of the two objects.
G is the Universal Gravitational Constant, a tiny number: 6.67 x 10⁻¹¹ N·m²/kg². The smallness of G is why gravity only feels strong when huge masses are involved.

Also, notice that r is squared in the denominator. This is an inverse square law. It means that if you double the distance between two objects, the gravitational force between them doesn't get cut in half. It gets cut to one-fourth (1/2²). If you triple the distance, the force drops to one-ninth (1/3²). This is why gravity gets much weaker as you get farther away from a planet.

The force is always attractive and always acts along the line connecting the two centers of mass. The Earth pulls on you, and you pull on the Earth with an equal and opposite force (thanks, Newton's Third Law!).

Gravitational Fields: A Planet's "Aura"

It can be strange to think about forces acting across empty space. Physicists use the concept of a field to help. Imagine a massive object like the Earth creates a "gravitational field" in the space around it. It's like an invisible region of influence. Any other mass that enters this field will feel a force.

We define the strength of the gravitational field, g, as the force per unit mass:

|g⃗| = |F⃗_g| / m

The units for g are newtons per kilogram (N/kg). If we substitute our universal gravitation equation for F_g, we get a useful new equation for the field created by a mass M:

|g⃗| = G * M / r²

This tells you the strength of the gravitational field at any distance r from the center of a planet of mass M.

Weight vs. Mass: An Important Distinction

Near the surface of the Earth, the value of g is pretty constant, about 9.8 N/kg. This value is so important it gets its own name: the acceleration due to gravity. And the gravitational force on an object near a planet's surface gets its own name: weight.

Weight = F_g = m * g

Mass is the amount of "stuff" in you (measured in kg). It's an intrinsic property. Your mass is the same whether you're in Dallas, on the Moon, or floating in deep space.

Weight is the force of gravity on that "stuff" (measured in N). Your weight depends on where you are. On the Moon, where g is about 1/6th of Earth's, you would weigh 1/6th as much.

For AP Physics 1, we often approximate g ≈ 10 m/s² or 10 N/kg to make the math easier. Notice the units are equivalent! An acceleration of 10 meters per second squared is numerically equal to a field strength of 10 newtons per kilogram.

Apparent Weight: The Elevator Problem

Your weight is the force of gravity, mg. But the "weight" you feel is actually the normal force from the floor pushing up on you. We call this your apparent weight.

Imagine you're standing on a bathroom scale in an elevator.

At rest or moving at a constant velocity
Your acceleration a = 0. The net force is zero. So, ΣF = F_N - mg = 0. The normal force equals your weight: F_N = mg. The scale reads your true weight.
Accelerating upward
You're moving up with acceleration a. The net force must be upward. ΣF = F_N - mg = ma. Solving for the normal force gives F_N = mg + ma. You feel heavier, and the scale shows a higher number.
Accelerating downward
You're moving down with acceleration a. The net force is downward. We'll make down the positive direction for a moment: ΣF = mg - F_N = ma. Solving for the normal force gives F_N = mg - ma. You feel lighter, and the scale shows a lower number.
Freefall
The elevator cable snaps! You and the elevator are both accelerating downward at a = g. Your apparent weight is F_N = mg - m(g) = 0. You are "weightless"! The scale reads zero. This is exactly what astronauts experience in orbit—they are constantly in freefall around the Earth.

Inertial vs. Gravitational Mass

This last point is a bit mind-bending, but it's a cornerstone of modern physics. We've actually been talking about two types of mass.

1
Inertial Mass
This is the m in F = ma. It's a measure of an object's inertia, or its resistance to being accelerated. It's how hard you have to push a stalled car to get it moving.
2
Gravitational Mass
This is the m in F_g = G(m₁m₂)/r². It's a measure of how much gravitational force an object exerts and feels. It's the "charge" for the gravitational force.

Why should the property that determines how hard something is to push be the exact same property that determines how strongly it's pulled by gravity? There's no obvious reason. Yet, countless experiments have shown that inertial mass and gravitational mass are equivalent. This profound idea is called the equivalence principle, and it's what led Albert Einstein to his theory of general relativity. For our purposes, it means we can just use m for both without worry.

Worked examples

Example 1

Force Between Two Bowling Balls

Problem: Two standard bowling balls, each with a mass of 7.0 kg, are sitting on a rack such that their centers are 0.5 meters apart. What is the gravitational force of attraction between them?

Solution:

1
Identify the Goal
We need to find the gravitational force, F_g, between the two bowling balls.
2
Choose the Right Tool
Since we have two masses and the distance between their centers, we'll use Newton's Law of Universal Gravitation: F_g = G(m₁m₂)/r².
3
List Your Knowns
- G = 6.67 x 10⁻¹¹ N·m²/kg²
- m₁ = 7.0 kg
- m₂ = 7.0 kg
- r = 0.5 m
4
Plug and Chug
Now, we substitute the values into the equation. F_g = (6.67 x 10⁻¹¹) * (7.0 * 7.0) / (0.5)² F_g = (6.67 x 10⁻¹¹) * (49) / (0.25) F_g = (6.67 x 10⁻¹¹) * 196 F_g ≈ 1.3 x 10⁻⁸ N

Why this matters: This force is about 13 nanonewtons. That is an astonishingly small force! It's roughly the weight of a single bacterium. This shows why you don't notice the gravitational pull from everyday objects. The force is there, but it's completely negligible compared to the pull of the Earth.

Example 2

Apparent Weight in an Elevator

Problem: A student, Priya, has a mass of 60 kg. She stands on a scale in an elevator in a tall building in Chicago. The elevator begins to accelerate upward from rest at a rate of 1.5 m/s². What does the scale read in newtons? (Use g = 10 m/s²).

Solution:

1
Identify the Goal
The scale reading is the apparent weight, which is the magnitude of the normal force, F_N.
2
Draw a Free-Body Diagram
Always start with a diagram! Draw a dot for Priya. Draw an arrow pointing down labeled F_g = mg. Draw an arrow pointing up labeled F_N. Since the elevator is accelerating up, the F_N arrow should be longer than the F_g arrow.
3
Choose the Right Tool
We'll use Newton's Second Law, ΣF = ma. The net force is the sum of the forces acting on Priya.
4
Set up the Equation
We'll define "up" as the positive direction. ΣF_y = F_N - F_g = ma F_N - mg = ma
5
Solve for the Unknown
We want to find F_N. Let's rearrange the equation first. F_N = mg + ma
6
Plug and Chug
Now substitute the values.
- m = 60 kg
- g = 10 m/s²
- a = 1.5 m/s² F_N = (60 kg)(10 m/s²) + (60 kg)(1.5 m/s²) F_N = 600 N + 90 N F_N = 690 N

Why this matters: Priya's true weight is mg = 600 N. But because the elevator is accelerating upward, the floor has to push on her with an extra 90 N of force to cause that acceleration. This extra push makes her feel heavier, and the scale reads 690 N. This is a classic AP Physics problem, so make sure you understand the logic!

Try it yourself

Problem 1

The planet Mars has a mass of 6.42 x 10²³ kg and a radius of 3.39 x 10⁶ m. What is the strength of the gravitational field (g) on the surface of Mars? How much would a 70 kg astronaut weigh on Mars?

Problem 2

Aaliyah is in an elevator and standing on a scale. Her mass is 55 kg. As the elevator nears the top floor, it slows down, accelerating at a rate of -2.0 m/s² (the negative sign means the acceleration is downward). What does the scale read? (Use g = 10 m/s²).

TL;DR

In simple terms, gravitational force is the universal attraction between any two objects with mass, from apples and planets to you and your pencil.

Syntax pattern

|F⃗_g| = G * (m₁ * m₂) / r²

Key terms

AP Physics 1 Gravitational force Newton's law of universal gravitation Weight vs mass Apparent weight Gravitational field Inverse square law Elevator problem Freefall Equivalence principle

You can now…

2.6.A: Describe the gravitational interaction between two objects or systems with mass.
2.6.B: Describe situations in which the gravitational force can be considered constant.
2.6.C: Describe the conditions under which the magnitude of a system's apparent weight is different from the magnitude of the gravitational force exerted on that system.
2.6.D: Describe inertial and gravitational mass.

Essential knowledge (exam-tested)

2.6.A.1: Newton's law of universal gravitation describes the gravitational force between two objects or systems as directly proportional to each of their masses and inversely proportional to the square of the distance between the systems' centers of mass. Relevant equation: |F⃗_g| = G(m₁m₂)/r²
2.6.A.1.i: The gravitational force is attractive.
2.6.A.1.ii: The gravitational force is always exerted along the line connecting the centers of mass of the two interacting systems.
2.6.A.1.iii: The gravitational force on a system can be considered to be exerted on the system's center of mass.
2.6.A.2: A field models the effects of a noncontact force exerted on an object at various positions in space.
2.6.A.2.i: The magnitude of the gravitational field created by a system of mass M at a point in space is equal to the ratio of the gravitational force exerted by the system on a test object of mass m to the mass of the test object.
2.6.A.2.ii: Derived equation: |g⃗| = |F⃗_g|/m = G(M/r²) If the gravitational force is the only force exerted on an object, the observed acceleration of the object (in m/s²) is numerically equal to the magnitude of the gravitational field strength (in N/kg) at that location.
2.6.A.3: The gravitational force exerted by an astronomical body on a relatively small nearby object is called weight. Derived Equation: Weight = F_g = mg
2.6.B.1: If the gravitational force between two systems' centers of mass has a negligible change as the relative position of the two systems changes, the gravitational force can be considered constant at all points between the initial and final positions of the systems.
2.6.B.2: Near the surface of Earth, the strength of the gravitational field is g ≈ 10 N/kg.
2.6.C.1: The magnitude of the apparent weight of a system is the magnitude of the normal force exerted on the system.
2.6.C.2: If the system is accelerating, the apparent weight of the system is not equal to the magnitude of the gravitational force exerted on the system.
2.6.C.3: A system appears weightless when there are no forces exerted on the system or when the force of gravity is the only force exerted on the system.
2.6.C.4: The equivalence principle states that an observer in a noninertial reference frame is unable to distinguish between an object's apparent weight and the gravitational force exerted on the object by a gravitational field.
2.6.D.1: Objects have inertial mass, or inertia, a property that determines how much an object's motion resists changes when interacting with another object.
2.6.D.2: Gravitational mass is related to the force of attraction between two systems with mass.
2.6.D.3: Inertial mass and gravitational mass have been experimentally verified to be equivalent.

Concept map

flowchart TD
    A[Any two objects with mass m1, m2] --> B{Newton's Law of Universal Gravitation};
    B --> C["F_g = G(m1 * m2) / r^2"];
    C --> D{Is one mass a huge planet, M?};
    D -- Yes --> E[Creates a Gravitational Field];
    E --> F["g = GM / r^2"];
    F --> G{Is a small object 'm' in the field?};
    G -- Yes --> H[Object has Weight];
    H --> I["F_g = mg"];
    I --> J{Is the object accelerating vertically?};
    J -- Yes --> K[Apparent Weight != True Weight];
    K --> L["F_normal = mg + ma"];
    D -- No --> M[Force is usually negligible];

Read what Saavi narrates

Have you ever wondered why the Moon doesn't just fly off into space? Or why it doesn't crash into the Earth? The same force that makes an apple fall from a tree is what holds the Moon in its orbit. It's gravity. But it’s not just for planets and apples. It's a universal attraction between any two things that have mass. You, your chair, your phone... everything is pulling on everything else.

In this lesson, we're going to explore this universal law. We'll see how it governs the orbits of planets and how it simplifies for everyday situations, like calculating your weight. We'll also tackle some classic physics scenarios, like what happens to your weight in an accelerating elevator.

Let's walk through an example. Imagine a student, Priya, has a mass of 60 kilograms. She stands on a scale in an elevator that starts accelerating upward at 1.5 meters per second squared. What does the scale read?

Well, her true weight is her mass times the acceleration of gravity... so 60 kilograms times about 10 meters per second squared, which is 600 Newtons. But the scale doesn't measure weight, it measures the normal force... the force of the floor pushing up.

Because the elevator is accelerating up, the net force must be up. So the normal force has to be bigger than her weight. We use Newton's Second Law: Net Force equals m a. That's Normal Force minus weight equals m a.
If we rearrange that, we find the Normal Force equals her weight, m g, plus m a.
Plugging in the numbers, we get 600 Newtons plus... 60 times 1.5... which is 90 Newtons. So the scale reads 690 Newtons. She feels heavier!

A really common mistake I see is students thinking that astronauts in orbit are weightless because there's no gravity in space. That's not true! The space station is close enough to Earth that gravity is still about 90 percent as strong. The "weightlessness" they feel is because they, and their spaceship, are in a constant state of freefall around the Earth.

So, as you work through these problems, remember that gravity is universal, and what you feel as weight can change. You've got this.

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Settings & Accessibility

Gravitational Force

Why this matters

Diagram

Concept map

Core explanation

The Universe's Great Attraction

Gravitational Fields: A Planet's "Aura"

Weight vs. Mass: An Important Distinction

Apparent Weight: The Elevator Problem

Inertial vs. Gravitational Mass

Worked examples

Force Between Two Bowling Balls

Apparent Weight in an Elevator

Try it yourself

Problem 1

Problem 2

Practice — 8 questions