Gravitational Force | AP Physics 1 Unit 2 Study Guide

Getting Started

Every object with mass in the universe pulls on every other object with mass. This fundamental interaction, known as gravity, governs everything from a dropped apple accelerating toward the ground to the orbit of planets around the sun. This chapter explores the nature of this universal force, examining how to calculate its strength and under what conditions we can simplify our model of it.

What You Should Be Able to Do

After completing this chapter, you will be able to:

Calculate the gravitational force between any two objects using Newton's law of universal gravitation.
Explain the concept of a gravitational field as a way to model this non-contact force.
Determine when it is appropriate to use the simplified, constant model for gravitational force (mg).
Distinguish between an object's true weight and its apparent weight, particularly in accelerating scenarios.
Compare and contrast the concepts of inertial mass and gravitational mass.

Key Concepts & Mechanisms

System & Preconditions

The system we consider for gravity is fundamentally composed of two or more objects possessing mass. The interaction is the mutual, attractive force between them. For calculations, we make a key idealization: we treat extended objects (like planets) as if all their mass were concentrated at a single point called the center of mass. This allows us to use a single, well-defined distance between the objects in our equations.

Key Steps / Relations

Newton's Law of Universal Gravitation: This law provides the magnitude of the attractive force between any two objects. The force is directly proportional to the product of the two masses and inversely proportional to the square of the distance separating their centers of mass.
The equation is:
$F_{g} = G \frac{m _{1} m _{2}}{r ^{2}}$
- $F_{g}$ is the magnitude of the gravitational force (in Newtons, N).
- $m_{1}$ and $m_{2}$ are the masses of the two objects (in kilograms, kg).
- $r$ is the distance between the centers of mass of the two objects (in meters, m).
- $G$ is the universal gravitational constant, an experimentally determined value approximately equal to $6.67 \times 1 0^{- 11} N \cdot m^{2} / kg^{2}$ . Its small value indicates that gravity is a very weak force unless at least one of the masses is astronomically large.
The Gravitational Field: To explain how gravity acts over empty space, physicists use the concept of a field. Any object with mass creates a gravitational field in the space around it. This field is a vector quantity that describes the gravitational force that would be exerted per unit of mass at any given point.
The gravitational field strength, g, is defined as:
$g = \frac{F _{g}}{m}$
- $g$ has units of Newtons per kilogram (N/kg), which is equivalent to meters per second squared (m/s²). Near Earth's surface, its value is approximately 9.8 N/kg.
Weight as a Gravitational Force: The force we call weight is simply the gravitational force exerted by a very large astronomical body (like a planet) on a smaller object near its surface. By combining the universal law and the definition of the field, we can see that the familiar equation for weight, $F_{g} = m g$ , is a special case of the universal law. For an object of mass m near Earth (mass $M_{E}$ , radius $R_{E}$ ), its weight is $F_{g} = G \frac{M _{E} m}{R _{E}^{2}}$ . The term $G \frac{M _{E}}{R _{E}^{2}}$ is the value of the gravitational field g at the surface.
Apparent Weight: While an object's true weight (mg) is the gravitational force on it, its apparent weight is the force the object exerts on its support. In most cases, this is equal to the magnitude of the normal force ( $F_{N}$ or $N$ ) exerted by the support on the object. If an object is in equilibrium (at rest or moving with constant velocity), its apparent weight equals its true weight. However, if the object is accelerating vertically, its apparent weight will change. For an object on a scale in an elevator:
- Accelerating Upward: The normal force must be greater than the weight to provide a net upward force ( $F_{N} > m g$ ). You feel heavier.
- Accelerating Downward: The normal force must be less than the weight to allow a net downward force ( $F_{N} < m g$ ). You feel lighter.
- In Freefall: The acceleration is equal to g, and the normal force is zero. The apparent weight is zero, a state often called "weightlessness."

Outputs & Effects

The primary effect of the gravitational interaction is the creation of a pair of attractive forces, one on each object. If this force is unbalanced, it will cause the object to accelerate according to Newton's second law. This acceleration is responsible for everything from falling objects to the orbits of moons and planets.

Regulation & Limits

The universal law $F_{g} = G \frac{m _{1} m _{2}}{r ^{2}}$ is always applicable. However, the simplified model $F_{g} = m g$ is an approximation that is valid only when the gravitational field can be considered constant. This holds true when the change in an object's distance from the center of the planet is negligible compared to the planet's radius. For example, for objects moving a few hundred meters up or down from Earth's surface, the value of g changes by a minuscule amount, so we can treat it as a constant 9.8 m/s².

Inertial vs. Gravitational Mass

Mass appears in two different fundamental contexts in physics. It is crucial to distinguish them, even though they are numerically equivalent.

Feature	Inertial Mass ( $m_{i}$ )	Gravitational Mass ( $m_{g}$ )	Why It Matters
Definition	A measure of an object's resistance to acceleration. It is the `m` in Newton's second law, $Σ F = m_{i} a$ .	A measure of the strength of the gravitational force an object exerts and experiences. It is the `m` in the law of universal gravitation, $F_{g} = G \frac{m _{g 1} m _{g 2}}{r ^{2}}$ .	These are conceptually distinct properties. One relates to inertia and changes in motion, while the other relates to a fundamental force.
How to Measure	Apply a known net force to an object and measure its resulting acceleration. Calculate $m_{i} = Σ F / a$ .	Place the object in a known gravitational field (or near another known mass) and measure the resulting gravitational force (e.g., with a spring scale). Calculate $m_{g} = F_{g} / g$ .	The measurement procedures are completely different, reflecting the different physical principles involved.
Equivalence	Numerous high-precision experiments have confirmed that inertial mass and gravitational mass are equivalent to an extraordinary degree of accuracy.	The Principle of Equivalence states that $m_{i} = m_{g}$ . This is a foundational assumption of Einstein's theory of general relativity.	This equivalence is why all objects in a vacuum, regardless of their mass, fall with the same acceleration. If $F_{g} = m_{g} g$ and $Σ F = m_{i} a$ , then $m_{g} g = m_{i} a$ . If $m_{g} = m_{i}$ , then $a = g$ .

Key Models & Diagrams

The model you choose for gravity depends on the physical situation. This table links common scenarios to the appropriate mathematical model and its key predictions.

Scenario	Appropriate Model / Equation(s)	Key Prediction or Observable
Calculating the force between Earth and the Moon.	Universal Gravitation: $F_{g} = G \frac{m _{E} m _{M}}{r ^{2}}$	The force is significant due to the large masses and is responsible for keeping the Moon in orbit. The force decreases if the distance `r` increases.
A box sliding down a ramp near Earth's surface.	Constant Local Gravity: $F_{g} = m g$	The gravitational force (weight) on the box is constant in magnitude and direction (straight down) throughout its motion on the ramp.
A person standing on a scale in an elevator accelerating upward.	Newton's Second Law & Apparent Weight: $Σ F_{y} = m a_{y}$ $F_{N} - m g = m a_{y}$	The scale reading ( $F_{N}$ ) will be greater than the person's true weight (`mg`). The person's apparent weight increases.

Key Components & Evidence

Gravitational Force ( $F_{g}$ ): The fundamental, attractive force between any two objects with mass. Its SI unit is the Newton (N).
Mass (m): An intrinsic property of matter. It is the source of gravity (gravitational mass) and a measure of inertia (inertial mass). Its SI unit is the kilogram (kg).
Universal Gravitational Constant (G): A fundamental constant of nature that sets the scale for the strength of gravity. $G \approx 6.67 \times 1 0^{- 11} N \cdot m^{2} / kg^{2}$ .
Gravitational Field (g): A vector field that describes the gravitational force per unit mass at any point in space. Its SI unit is N/kg or m/s².
Weight: The common name for the gravitational force exerted by a planet or moon on a nearby object.
Apparent Weight: The contact force, typically the normal force, supporting an object. It can differ from true weight if the object is accelerating.
Center of Mass: The point at which an object's mass can be considered to be concentrated for the purpose of gravitational calculations.
Inverse Square Law: The principle that the strength of the gravitational force is inversely proportional to the square of the distance ( $F_{g} \propto 1/ r^{2}$ ).

Skill Snapshots

Causation:
- An object with mass creates a gravitational field that permeates the space around it.
- The interaction between a second object's mass and the existing field results in a gravitational force on that second object.
- A net vertical acceleration causes an object's apparent weight (the normal force) to be different from its true gravitational weight.
Comparison:
- The universal gravitation model ( $F_{g} \propto 1/ r^{2}$ ) is a precise model for all scales, whereas the local gravity model ( $F_{g} = m g$ ) is a convenient and accurate approximation only when an object's height changes negligibly compared to the radius of the planet.
- Inertial mass measures an object's resistance to a change in velocity, while gravitational mass determines the strength of the gravitational force it exerts and experiences.
- True weight (mg) is the constant gravitational pull on an object, while apparent weight (the normal force) is a contact force that can change if the object is accelerating.
Change Over Time:
- Baseline: An object at rest on a horizontal surface on Earth experiences a constant downward gravitational force mg.
- Change 1: If the object is launched into a high-altitude orbit, the distance r from Earth's center increases significantly, causing the gravitational force on it to decrease.
- Change 2: If the surface supporting the object (like an elevator floor) begins to accelerate downward, the required normal force decreases, causing the object's apparent weight to decrease.
- Continuity: Throughout these changes in position and motion, the object's inertial and gravitational mass remain constant.

Common Misconceptions & Clarifications

Misconception: "There is no gravity in space."
- Clarification: Gravity is a universal, long-range force. The International Space Station, for example, experiences about 90% of the gravitational force it would on Earth's surface. Astronauts appear "weightless" because they and their spacecraft are in a continuous state of freefall around the Earth.
Misconception: "Weight and mass are the same."
- Clarification: Mass is an intrinsic property of an object measuring its matter content and inertia. Weight is the gravitational force acting on that mass. An astronaut has the same mass on Earth and the Moon, but their weight on the Moon is about 1/6th of their weight on Earth because the Moon's gravitational field is weaker.
Misconception: "The Earth pulls on the Moon, but the Moon doesn't pull on the Earth."
- Clarification: According to Newton's third law, forces always come in interaction pairs. The force the Earth exerts on the Moon is exactly equal in magnitude and opposite in direction to the force the Moon exerts on the Earth. The Earth's resulting acceleration is much smaller only because its mass is much larger.
Misconception: "Heavier objects fall faster."
- Clarification: In the absence of air resistance, all objects fall with the same acceleration (g). This is a direct consequence of the equivalence of inertial and gravitational mass. A more massive object experiences a greater gravitational force, but it also has proportionally greater inertia, so the resulting acceleration is the same.

One-Paragraph Summary

The gravitational force is a universal, attractive interaction between any two objects with mass. Described by Newton's law of universal gravitation, its strength is proportional to the product of the masses and inversely proportional to the square of the distance between their centers. For objects near a planet's surface, this law simplifies to the familiar constant-force model of weight, $F_{g} = m g$ . It is crucial to distinguish this true weight from apparent weight, which is the normal force and can vary if the object is accelerating. Finally, the concept of mass itself is twofold: inertial mass resists changes in motion, while gravitational mass is the source of the gravitational force. The experimentally verified equivalence of these two types of mass is a cornerstone of modern physics.

Gravitational Force - AP Physics 1: Algebra-Based Study Guide