2.6 Gravitational Force

AP Physics 1 2.6 Gravitational Force Summary

Gravitational force is the attractive pull between any two objects with mass, given by $F_g = G\frac{m_1 m_2}{r^2}$. It always acts along the line connecting their centers of mass. Near Earth's surface you can treat the gravitational field as roughly constant ($g \approx 10$ N/kg), and your apparent weight (the normal force you feel) can differ from your true weight $mg$ whenever you accelerate vertically.

Why This Matters for the AP Physics 1 Exam

Gravitational force shows up across many AP Physics 1 problems, from free-fall motion to orbits and circular motion later in the course. You need to connect the equation to its meaning: the inverse-square relationship, the difference between mass and weight, and why apparent weight changes when there is acceleration.

This topic is strong material for the kind of free-response question that asks you to translate between words, diagrams, and equations. You might describe a scenario in words first, then derive an expression, then explain how your reasoning and your equation agree. Being able to start from a free-body diagram and $\sum F = ma$ instead of memorizing a separate "apparent weight" formula is exactly the reasoning that pays off on the exam.

Key Takeaways

• Gravitational force is attractive, acts along the line connecting two centers of mass, and follows the inverse-square law $F_g = G\frac{m_1 m_2}{r^2}$.
• Gravitational field strength is force per unit mass: $g = \frac{F_g}{m} = G\frac{M}{r^2}$, measured in N/kg. If gravity is the only force, acceleration in m/s² equals field strength in N/kg.
• Weight is a gravitational force ($F_g = mg$) and changes with location; mass stays the same everywhere.
• Near Earth's surface, gravity is approximately constant because elevation changes are tiny compared to Earth's radius. Use $g \approx 10$ N/kg on the exam.
• Apparent weight is the normal force. Derive it from a free-body diagram and $\sum F = ma$, not from a memorized formula.
• Inertial mass (resistance to acceleration) and gravitational mass (strength of gravitational attraction) are experimentally equivalent, which is why all objects fall at the same rate when air resistance is negligible.

Gravitational Interaction Between Objects

Newton's Law of Universal Gravitation Formula

Newton's law of universal gravitation describes how any two objects with mass attract each other. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers of mass:

$F_g = G \frac{m_1 m_2}{r^2}$

Where:

• $F_g$ is the gravitational force between the objects
• $G$ is the universal gravitational constant ($6.67 \times 10^{-11}$ N·m²/kg²)
• $m_1$ and $m_2$ are the masses of the two objects
• $r$ is the distance between the centers of mass of the objects

The two objects exert equal-magnitude, attractive forces on each other along the line connecting their centers of mass. For modeling translational motion, you can treat the gravitational force on each system as acting at that system's center of mass.

Because the equation depends on mass as a product and distance as an inverse square:

• Doubling both masses makes the force 4 times stronger.
• Doubling the distance makes the force 4 times weaker ($1/2^2 = 1/4$).

These are the key facts about gravitational force:

• It always acts along the line connecting the two centers of mass.
• It can be treated as acting at an object's center of mass, regardless of shape.
• It is always attractive, never repulsive.
• It applies to any two objects or systems with mass.

Application: The Moon orbits Earth, and Earth orbits the Sun, because of this gravitational attraction.

Gravitational Field Model

A field models the effect of a noncontact force at different positions in space. Instead of only thinking about two objects pulling on each other directly, you can picture a gravitational field that any mass creates around itself.

The gravitational field strength at a point is the gravitational force on a test object divided by that object's mass:

$g = \frac{F_g}{m} = G\frac{M}{r^2}$

Where:

• $g$ is the gravitational field strength (in N/kg)
• $G$ is the universal gravitational constant
• $M$ is the mass creating the field
• $r$ is the distance from the center of mass

So field strength is force per unit mass. If the gravitational force is the only force on an object, the object's acceleration in m/s² is numerically equal to the field strength in N/kg at that location. This is why all objects fall at the same rate in a vacuum, regardless of mass.

Example: On Earth's surface, the field strength is about 9.8 N/kg, so an object released from rest accelerates downward at about 9.8 m/s².

Weight as Gravitational Force

Weight is the gravitational force that an astronomical body exerts on a relatively small nearby object. Mass is an intrinsic property of matter and does not change with location. Weight is a force and changes with the local field strength.

$Weight = F_g = mg$

Where:

• $m$ is the object's mass (in kilograms)
• $g$ is the local gravitational field strength (in N/kg)

Key points about weight:

• Weight is directly proportional to mass in a given field, so doubling the mass doubles the weight.
• A 1 kg object weighs about 9.8 N on Earth but only about 1.6 N on the Moon, because the field strengths differ.
• Weight is measured in newtons; mass is measured in kilograms.

This is why astronauts in orbit feel weightless even though they still have the same mass and still experience a large gravitational force. Gravity is the only force acting on them, so their apparent weight (the normal force) is zero.

When Gravitational Force Can Be Treated as Constant

If the distance between two systems changes by only a negligible amount during the motion, the gravitational force between them can be treated as constant over that interval. This approximation makes motion near Earth's surface much easier to analyze.

The field strength at Earth's surface is about 9.8 N/kg, often rounded to 10 N/kg on the AP Physics 1 exam. There are small variations with latitude and elevation, but for most everyday situations they do not matter.

This approximation works because:

• Changes in elevation near Earth's surface are tiny compared to Earth's radius (about 6,371 km).
• For buildings, airplanes, and lab experiments, the change in gravity is negligible.
• The approximation breaks down far from Earth's surface, such as for satellites in high orbit.

Example: A 5 kg object weighs about 49 N anywhere on Earth's surface, with only small variations.

Apparent Weight vs Gravitational Force

Normal Force and Apparent Weight

What you feel as "weight" in daily life is your apparent weight, and it can differ from the actual gravitational force on you. Apparent weight is the magnitude of the normal force exerted on the system, such as the force from a scale or floor.

Start from a free-body diagram and Newton's second law. For a person on a scale with vertical acceleration $a$ (taking up as positive), the forces are the normal force $N$ up and gravity $mg$ down:

$\sum F = N - mg = ma$

Solving for the scale reading:

$N = mg + ma = m(g + a)$

When there is no vertical acceleration ($a = 0$), $N = mg$, so apparent weight equals true weight. When the object accelerates, $N$ changes, so apparent weight no longer equals the gravitational force.

This single relationship explains everything:

• A scale reads more than $mg$ when the acceleration is upward (positive $a$).
• A scale reads less than $mg$ when the acceleration is downward (negative $a$).
• In free fall, $a = -g$, so $N = m(g - g) = 0$ and the scale reads zero.

Acceleration Effects on Apparent Weight

Your mass and the gravitational force on you stay the same in an elevator, but your apparent weight changes because the normal force changes. Use the same relationship $N = m(g + a)$, where $a$ is positive when the acceleration points up and negative when it points down.

• Elevator accelerating upward: $a > 0$, so $N > mg$ and you feel heavier.
• Elevator accelerating downward: $a < 0$, so $N < mg$ and you feel lighter.
• Top of a roller coaster hill: the downward acceleration can make the normal force drop, so you feel briefly weightless.

Example: A 70 kg person in an elevator accelerating upward at 2 m/s² has an apparent weight of $N = 70(9.8 + 2) = 826$ N, more than the true weight of 686 N.

Weightlessness Conditions

A system appears weightless when no forces act on it, or when gravity is the only force acting on it. In both cases the apparent weight (the normal force) is zero.

Objects in orbit appear weightless because they are in continuous free fall. The spacecraft and the astronauts inside fall around Earth at the same rate, so the astronauts do not press against the spacecraft's surfaces and feel no normal force.

The feeling of weightlessness comes from the absence of a normal force on the body. You can experience it in:

• Spacecraft in orbit
• Objects in free fall
• Aircraft flying in parabolic arcs, which create brief periods of free fall

Equivalence Principle

The equivalence principle states that an observer in a noninertial reference frame cannot distinguish between an object's apparent weight and the gravitational force from a gravitational field. In a closed elevator accelerating upward at about 9.8 m/s², any experiment you ran would give the same results as standing at rest in Earth's gravitational field.

Inertial vs Gravitational Mass

Inertial Mass

Inertial mass measures how strongly an object resists changes in its motion when forces act on it. Objects with greater inertial mass need larger forces to reach the same acceleration, as described by Newton's second law:

$F = ma$

Where:

• $F$ is the net force applied
• $m$ is the inertial mass
• $a$ is the resulting acceleration

Inertial mass tells you how hard it is to start, stop, or turn an object, and it acts as the proportionality constant between force and acceleration.

Example: A bowling ball has more inertial mass than a basketball, so you need a larger force to give it the same acceleration.

Gravitational Mass

Gravitational mass describes how strongly an object interacts through gravity. It sets both the gravitational force an object exerts on others and the force it feels from others. It appears as the masses in the universal gravitation equation:

$F_g = G \frac{m_1 m_2}{r^2}$

Equivalence of Mass Types

Inertial mass and gravitational mass have been experimentally verified to be equivalent. That equivalence is why all objects in the same gravitational field have the same free-fall acceleration when air resistance is negligible. Whether you measure an object's mass by how it accelerates under a known force or by the gravitational force it feels in a known field, you get the same value.

How to Use This on the AP Physics 1 Exam

Problem Solving

• For two-body gravitation problems, plug directly into $F_g = G\frac{m_1 m_2}{r^2}$ and keep track of powers of ten. Watch the units on $G$.
• Use the inverse-square relationship to reason about changes without recalculating. If $r$ triples, $F_g$ drops by a factor of 9.
• For field problems, remember $g = G\frac{M}{r^2}$ gives field strength in N/kg, which is numerically equal to free-fall acceleration in m/s² when gravity is the only force.

Free Response

• For apparent weight, always draw a free-body diagram first, write $\sum F = ma$, then solve for the normal force. Do not start from a memorized "$mg \pm ma$" formula.
• When a question asks for a description, claim, and then a derivation, make sure your final explanation connects your equation back to your words. State whether the scale reads more or less than $mg$ and why.
• Be explicit about sign conventions. Pick up as positive, then let $a$ be negative for downward acceleration.

Common Trap

• Do not confuse mass and weight. Mass is in kilograms and never changes with location; weight is in newtons and depends on $g$.
• Apparent weight is the normal force, not the gravitational force. They are equal only when there is no vertical acceleration.

Common Misconceptions

• "Astronauts in orbit are weightless because they are beyond gravity's reach." They still experience a large gravitational force. They feel weightless because they and their spacecraft fall around Earth at the same rate, so the normal force on them is zero.
• "Heavier objects fall faster." When air resistance is negligible, all objects in the same field accelerate at the same rate because inertial and gravitational mass are equivalent.
• "Apparent weight is a separate formula, $F_{apparent} = mg \pm ma$." Apparent weight is the normal force you get from a free-body diagram and $\sum F = ma$. The sign comes from your chosen coordinate direction, not from memorizing two formulas.
• "Mass and weight are the same thing." Mass measures the amount of matter and resistance to acceleration; weight is the gravitational force on that mass and changes with location.
• "Doubling the distance halves the gravitational force." Because of the inverse-square law, doubling the distance makes the force one-quarter as strong, not one-half.
• "The gravitational force points straight down everywhere." It points along the line connecting the two centers of mass. Near Earth's surface that happens to be very close to straight down, but the general rule is the line of centers.

Related AP Physics 1 Guides

• 2.1 Systems and Center of Mass
• 2.2 Forces and Free-Body Diagrams
• 2.3 Newton's Third Law
• 2.4 Newton's First Law
• 2.5 Newton's Second Law
• 2.7 Kinetic and Static Friction