Gravitational Force
Newtonian gravity is an attractive inverse-square interaction between masses. Near Earth’s surface it is often approximated by the nearly uniform force mg, but for large altitude changes the full GMm/r² dependence must be retained.
Topic sequence · 6 of 10
Learning objectives
Objective 1
Apply the inverse-square gravitational force law with correct vector direction.
Objective 2
Distinguish local gravitational field strength g from the universal constant G.
Objective 3
Decide when the near-surface mg approximation is adequate.
- Vector direction and unit-vector notation.
- Newton’s second law and circular-motion preparation.
- Distance measured from the center of a spherically symmetric gravitating body.
Definitions first
Representations you must read and create
Representation 1
Radial field-vector diagram around a spherical mass.
Representation 2
Graph of |F_g| or g versus radius showing 1/r² dependence.
Representation 3
Free-body diagram for orbit, falling object, or apparent-weight situation.
Representation 4
Graph of gravitational field/force versus radius showing linear behavior inside a uniform sphere and inverse-square behavior outside.
Representation 5
Elevator/free-fall free-body diagrams distinguishing true gravitational force from scale reading (apparent weight).
Core relationships and calculus
Variables: r from source center; r̂ outward
Use: gravitational force from spherical source
Variables: g in N/kg or m/s²
Use: gravitational field outside spherical source
Variables: near-surface uniform-field approximation
Use: weight over small altitude range
Variables: support/normal force magnitude
Use: apparent weight measured by a scale
Variables: uniform sphere density ρ; interior radius r
Use: mass enclosed within radius r
Variables: uniform sphere of total mass M and radius R; r<R
Use: gravitational field inside a uniform solid sphere
Variables: test mass m inside uniform sphere
Use: interior gravitational force proportional to r
Gravity produces position-dependent acceleration d²r⃗/dt²=−GM r⃗/r³. For radial or orbital motion this equation is not generally a constant-acceleration problem. Later energy and angular-momentum methods often provide more efficient solutions.
Outside a spherically symmetric source, Newtonian gravity has the same form as if the source mass were concentrated at its center. Substituting F_g into m a cancels the test mass, giving g=GM/r².
Use Newton’s shell theorem as a result; the current Mechanics scope does not require a mathematical proof or derivation of the shell theorem.
Physical examples
Example 1
Doubling distance from a planet center reduces gravitational force to one quarter if both masses are unchanged.
Example 2
Astronauts in orbit are not beyond gravity; they continuously accelerate toward Earth while falling around it.
Example 3
For a laboratory-scale vertical displacement near Earth, the fractional change in r is tiny, so treating g as constant is usually excellent.
Example 4
Inside a thin uniform spherical shell, contributions from the entire shell cancel to zero net gravitational force; outside, the shell acts as if all its mass were concentrated at the center.
Example 5
In free fall, gravity can be nonzero while apparent weight is zero because no support force acts on the object.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Measure r from the source center in the inverse-square law; altitude is only part of that distance.
Reasoning pattern 2
Use vector direction separately from magnitude so attraction is not lost in scalar algebra.
Reasoning pattern 3
When comparing two radii, ratios such as g_2/g_1=(r_1/r_2)^2 often reveal the physics faster than inserting G numerically.
Reasoning pattern 4
For a point inside a uniform solid sphere, use only the enclosed mass in the inverse-square relation; the outer spherical shells contribute zero net force.
Reasoning pattern 5
Do not equate apparent weight with mg automatically; apparent weight is the support-force magnitude.
Deeper calculus and limiting-case check
For a spherical source, g(r)=GM/r² implies dg/dr=−2GM/r³ outside the body, quantifying how rapidly the field changes with radius. Near Earth’s surface over small Δr, the fractional change is roughly 2Δr/R_E, explaining why constant g is such a strong local approximation. Inside a uniform sphere, M_enc∝r³ while the inverse-square law contributes 1/r², so the net field is proportional to r; this limiting behavior must join continuously to the exterior 1/r² field at the surface.
Model-selection workflow: decide what the problem is before choosing an equation
For Gravitational Force, the fastest reliable route is not equation hunting. Start by identifying the system and interaction, encode it with the free-body diagram, choose the Newton-law or momentum-rate equation, and then test the inertial-frame and force-model assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat gravitational force and universal gravitational constant as physical quantities first and symbols second.
Choose a representation
Build Radial field-vector diagram around a spherical mass.. If the prompt supplies Graph of |F_g| or g versus radius showing 1/r² dependence., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is F⃗_g = −GMm/r² r̂. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Use Newton’s shell theorem as a result; the current Mechanics scope does not require a mathematical proof or derivation of the shell theorem. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Radial field-vector diagram around a spherical mass. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | F⃗_g = −GMm/r² r̂ | gravitational force from spherical source. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | g⃗ = −GM/r² r̂ | gravitational field outside spherical source. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Gravity produces position-dependent acceleration d²r⃗/dt²=−GM r⃗/r³. For radial or orbital motion this equation is not generally a constant-acceleration problem. Later energy and angular-momentum methods often provide more efficient solutions. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A satellite of mass m orbits at radius r around Earth. Its gravitational acceleration magnitude is GM_E/r². Because satellite mass cancels from F=ma, all small satellites at the same orbital radius have the same gravitational acceleration independent of their mass. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 2.6
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship F⃗_g = −GMm/r² r̂ should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Radial field-vector diagram around a spherical mass. and Graph of |F_g| or g versus radius showing 1/r² dependence.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Measure local g with a pendulum or free-fall sensor, then compare with the gravitational-field model. Discuss why the experiment measures the local near-surface value rather than G directly.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Using altitude above the surface as r in GMm/r² instead of distance from the center. | Return to the free-body diagram, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Confusing G with g. | Return to the Newton-law or momentum-rate equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming gravitational force disappears in orbit. | Return to the free-body diagram, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 2.10 uses gravity as the centripetal interaction for orbital and loop examples. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Unit 3 introduces gravitational potential energy U=−GMm/r. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Unit 6 returns to satellite motion using energy and angular momentum. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write F⃗_g = −GMm/r² r̂ and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 2.6 mastery ladder
Experimental connection
Measure local g with a pendulum or free-fall sensor, then compare with the gravitational-field model. Discuss why the experiment measures the local near-surface value rather than G directly.
Connections across Physics C
Connection 1
Topic 2.10 uses gravity as the centripetal interaction for orbital and loop examples.
Connection 2
Unit 3 introduces gravitational potential energy U=−GMm/r.
Connection 3
Unit 6 returns to satellite motion using energy and angular momentum.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Using altitude above the surface as r in GMm/r² instead of distance from the center.
Avoid this
Confusing G with g.
Avoid this
Assuming gravitational force disappears in orbit.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Why does satellite mass cancel from acceleration?
Both gravitational force and inertial response scale with the satellite mass, so the ratio F/m is independent of it.
When can I use mg?
When the object remains in a region small compared with the source-body radius so g changes negligibly.
Is apparent weight the same as gravity?
No. Apparent weight usually refers to a support or normal-force reading, which can differ from mg during acceleration.