Unit 7 · Topic 02

Gravitational field and weight

The field characterizes the gravitational environment; a body's weight depends on that local field.

Unit 7Field and weightOpen navigation

Concept 01

Force per unit mass

Essential The minimum you should retain

The field is g=F/m and outside a spherical mass it is g=GM/r2 g=GM/r^2 toward the center.

UnderstandInterpret and connect

g is a field acceleration, not itself a force.

DeepenFormulation and conditions

An ideal test mass does not appreciably alter the source.

ExploreConnections for further study

Field maps compare environments before a test object is chosen.

Mathematical relation

Gravitational field and weight

g=GMr2,W=mg,g(h)=g0R/(R+h)2 g=GM/r^2,\quad W=mg,\quad g(h)=g_0[R/(R+h)]^2
Represents

Exterior field, weight, and radial variation.

Physical interpretation

g is not universal, and weight depends on the local field.

DeepenVariables, conditions, and checks

Variables

g
gravitational field or acceleration; usual unit: m/s²
W
gravitational weight; usual unit: N

Conditions of application

  • Spherically symmetric source; h is measured from the surface.

Dimensional check

m/s² and N.

normalized gravitational field

Normalized g curve versus altitude measured in radii.h=0h=Rh=2R
  • g/g₀

g decreases continuously with altitude and remains nonzero at h=R and h=2R.

Concept 02

Mass and weight

Essential The minimum you should retain

In a local field g, the gravitational weight of mass m is W=mg.

UnderstandInterpret and connect

Mass remains the same; weight changes with g.

DeepenFormulation and conditions

W=mg uses the field evaluated across the region occupied by the body.

ExploreConnections for further study

A scale measures a contact force, so its reading is apparent weight.

Worked example

Weight at one and a half Earth radii

A 70 kg person is at r=1.5R_E and g₀=9.8 m/s².

Given
  • m=70 kg
  • r=1.5R_E
  • g₀=9.8 m/s²
Target

Find g and W.

  1. Ratio

    g/g₀=(R/r)²=1/(1.5)².

  2. Field

    g=9.8/2.25=4.36 m/s².

  3. Weight

    W=mg=(70)(4.36).

  4. Result

    W≈305 N.

Conclusion

g≈4.36 m/s² and W≈305 N.

Concept 03

Field with altitude

Essential The minimum you should retain

At height h above radius R, g(h)=g₀[R/(R+h)]².

UnderstandInterpret and connect

The decrease is continuous and distance is measured from the center.

DeepenFormulation and conditions

At r=2R the field is g₀/4.

ExploreConnections for further study

The field does not switch off at an orbital altitude.

Concept 04

Apparent weightlessness

Essential The minimum you should retain

A spacecraft and its occupants may fall together and have zero apparent weight while g is nonzero.

UnderstandInterpret and connect

The absence of a normal force explains weightlessness.

DeepenFormulation and conditions

An orbital path still has radial gravitational acceleration.

ExploreConnections for further study

Separating gravitational force from scale reading removes the apparent paradox.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Treating 9.8 m/s² as a universal g.

g depends on M and r.

Explaining orbital weightlessness by vanished gravity.

Spacecraft and occupants share free fall in a nonzero field.