Gravitational force and field strength
You will be able to: Use an inverse-square law, distinguish force from field and explain the constant-g approximation.
Why does gravity weaken as you move away from a planet?
A small spacecraft at twice a planet’s radius from its center experiences one-quarter of the surface gravitational field. Twice the altitude is not the same as twice the center-to-center distance.
A useful starting point: Choose useful axes on an incline →
Words and symbols before equations
- G
- Universal gravitational constant, approximately 6.67 × 10⁻¹¹ N·m²/kg².
- r
- Distance between centers for spherical bodies outside one another, in m.
- Field strength g
- Gravitational force per unit test mass, in N/kg, numerically equal to free-fall acceleration in m/s².
- Inertial and gravitational mass
- Mass in F = ma and mass governing gravitational interaction; experiments show their equivalence.
What this picture assumes
Spherical source with surface field 10 N/kg. Probe is at or outside the surface. r/R is center distance, not altitude; source properties stay fixed.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- At r = 2R: g = 2.5 N/kg. On 2 kg, force = 5 N toward the center. Geometry is schematic; use the labeled ratio.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
For point masses or nonoverlapping spherically symmetric bodies, force magnitude is GMm/r², directed attractively along the line of centers. Dividing by the test mass m gives field magnitude GM/r². The source mass determines the field; the test mass determines how much force it feels.
A larger falling mass experiences proportionally larger gravity and proportionally greater inertia. Their cancellation gives the same ideal free-fall acceleration. This explanation assumes gravity is the only significant force, not substantial air resistance.
Near a planet’s surface, heights small compared with its radius cause only a small change in r and therefore in g. Use constant g for such local problems. Farther away, use r = R + h and the inverse-square relation instead.
A worked example, step by step
A planet has surface field 10 N/kg. Find the field and force on a 2 kg object at height h = R above its surface.
- Distance from the center is r = R + h = 2R.
- The inverse-square factor is (R/2R)² = 1/4.
- Field strength is 2.5 N/kg, toward the center.
- Force on 2 kg is mg = 2(2.5) = 5 N, also toward the center.
Use distance from the planet’s center, not height above the surface, in GM/r².
Does doubling test mass double the gravitational field at its location?
Compare with an explanation
No. It doubles the force on the test object, while the source field remains the same.
Predict. Change one thing. Explain.
Increase r/R from 1 to 2 while keeping both masses fixed. Predict field and force ratios. Then change only the test mass and compare force with field.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At r = 2R: g = 2.5 N/kg. On 2 kg, force = 5 N toward the center. Geometry is schematic; use the labeled ratio.
Spherical source with surface field 10 N/kg. Probe is at or outside the surface. r/R is center distance, not altitude; source properties stay fixed.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, system boundary, acceleration or calculus relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA source has surface field g₀. At r = 3R, find field and force on mass m. Explain why masses m and 2m have the same gravitational acceleration there.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: g = g₀/9, toward the center.
- 1 point: Force on m is mg₀/9.
- 1 point: Force on 2m is twice as large.
- 1 point: Dividing each force by the corresponding inertial mass gives the same acceleration g₀/9.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What distinguishes force from field?
Force depends on the test mass; field is force per unit test mass.
RECALL 2What distance enters the inverse-square law?
Center-to-center distance for the stated spherical or point-mass model.
RECALL 3Why can different masses fall together?
Gravitational force and inertia scale equally when other forces are negligible.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Gravitational force and field strength
- F_g = GMm/r²; g = GM/r² outside a spherical source.
- At altitude h: r = R + h.
- Inertial and gravitational mass scale equally in ideal free fall.
Remember: Use distance from the planet’s center, not height above the surface, in GM/r².
Conditions: Spherical source with surface field 10 N/kg. Probe is at or outside the surface. r/R is center distance, not altitude; source properties stay fixed.
Refresh Kid · AP Physics C: Mechanics Unit 2 (official Unit 2) · Objectives 2.6.A; 2.6.B; 2.6.D · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.6, objectives 2.6.A; 2.6.B; 2.6.D. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026 alongside the Fall 2026 clarifications. This is Mechanics Unit 2: Force and Translational Dynamics. The unit covers Topics 2.1–2.10. Calculus is introduced where it is needed for continuous mass and velocity-dependent forces. Shell theorem is applied without requiring a proof; spring combinations are purely series or purely parallel. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
Want to work through this with a tutor?
Bring your question about Gravitational force and field strength. Your explanation and answers remain free to access.