Distinguish gravitational field, mass and weight

1. Orient: Distinguish gravitational field, mass and weight

For two point masses separated by center-to-center distance r, the gravitational attraction has magnitude GMm/r². A spherically symmetric body produces the same external field as if its mass were concentrated at its center. The inverse-square expression here applies outside that body; it is not a general formula for the field inside an arbitrary distribution.

2. Explain and derive

Divide the force on a test object by its mass to define gravitational field strength g=GM/r². Its units N/kg are equivalent to m/s². Weight is the gravitational force mg, while mass is measured in kilograms and does not change merely because the object is carried to a different location.

Radius is measured from the central body’s center, not from its surface. If altitude equals one body radius, center distance is two radii and field strength is one quarter of the surface value. Doubling the central mass at the same radius doubles the field; doubling both mass and radius produces half the original field.

Gravity supplies an acceleration even during orbital free fall. An orbiting object is not beyond gravity simply because a support scale can read zero: a scale measures contact support, not gravitational force directly. The body and test object exert equal opposite gravitational forces, but their accelerations differ because their masses differ.

3. Work an example

The reference body has a surface field of 10 N/kg. At center distance 2 reference radii, the field is 10/4=2.5 N/kg. A 2 kg test object then has weight 5 N while its mass remains 2 kg. With central mass doubled at the same distance, the field is 5 N/kg and the test weight is 10 N.

4. Predict and investigate

Predict first: If altitude doubles from one body radius to two body radii, does weight fall by a factor of four? Convert altitude to center distance first.

Compare center distances 1,2 and 3 reference radii at fixed central mass. Then double central mass at fixed radius. Test the ratios without recalculating G. Record field and weight separately. Explain why the altitude column is always one less than the center-distance ratio. Use the lesson controls and record the settings with your evidence.

5. Transfer and explain

Try a new practice case

A 2 kg object is at a location with gravitational field 3 N/kg. Find its weight.

Your numerical result (N) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

Which quantity would a support scale measure in a freely falling elevator, and why need that reading not equal the gravitational force?

Retain your first answer, force diagram, equations, units, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Supplied synthetic outputs do not certify actual laboratory skill.

Local gravity, apparent weight and the two mass roles

A constant-g approximation is justified when the change in center distance is small enough that the inverse-square field variation is negligible for the required accuracy. At altitude h=0.01R above a spherical body, g/g_surface=1/(1.01)²≈0.980296, a decrease of about 1.97 percent; at h=R the decrease is 75 percent. The same approximation cannot be accepted without an accuracy criterion for both intervals.

In an upward-positive accelerating elevator, N−mg=ma_y, so a support scale reads N=m(g+a_y) while contact persists. For m=3 kg,g=10 m/s², upward acceleration 2 m/s² gives 36 N, downward acceleration 2 m/s² gives 24 N, and free fall gives zero. Horizontal acceleration alone need not change vertical support: the source’s statement about acceleration and apparent weight must be interpreted through the actual force direction, not as a universal assertion for every acceleration vector.

A local scale reading in a small closed laboratory cannot by itself distinguish a uniform gravitational field from a suitable accelerating frame; tidal variation or other external information can distinguish broader situations. This is the local equivalence-principle limit, not a claim that all gravitational fields are globally uniform. Inertial mass appears in Fnet=m_i a, while gravitational mass appears in Fg=m_g g_field. Their experimentally established equality makes free-fall acceleration independent of the test body’s mass. Changed check: a 2 kg body subject to a calibrated 6 N horizontal net force has inertial mass 6/3=2 kg if its acceleration is 3 m/s²; its gravitational mass inferred from weight 20 N in field 10 N/kg is also 2 kg. These are different measurement roles, not two new kinds of matter.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 2 topic mapping: 2.6. Checked 30 October 2026. This is a content cross-reference, not AP authorization.