AcademyGravitation
Academy
Weight in Gravitational Fields
Level 1 - Physics topic page in Gravitation.
Principle
Weight is the gravitational force produced by the local field, so it depends on both mass and location.
Notation
\(m\)
object mass
\(\mathrm{kg}\)
\(\vec W\)
weight force
\(\mathrm{N}\)
\(W\)
weight magnitude
\(\mathrm{N}\)
\(\vec g\)
local gravitational field
\(\mathrm{N\,kg^{-1}}\)
\(g\)
field magnitude
\(\mathrm{N\,kg^{-1}}\)
\(G\)
gravitational constant
\(\mathrm{N\,m^{2}\,kg^{-2}}\)
\(M\)
source mass
\(\mathrm{kg}\)
\(r\)
distance from the source center
\(\mathrm{m}\)
Method
Derivation 1: Define weight from the local field
Weight is a force, not a property stored in the object. Once the field is known at a position, the weight follows immediately from the object's mass.
Field form of gravity
\[\vec W=m\vec g\]
Weight magnitude
\[W=mg\]
The mass stays fixed while the local value of \(g\) can change.
Derivation 2: Build weight from the source mass
If the field comes from a spherical source, substitute the inverse-square field into the weight definition.
Spherical field magnitude
\[g=G\frac{M}{r^2}\]
Weight from source mass
\[W=G\frac{Mm}{r^2}\]
Near-Earth approximation
\[W\approx mg\quad\text{with}\quad g\approx9.8\,\mathrm{N\,kg^{-1}}\]
This is valid when the height change is tiny compared with Earth's radius.
The graph fixes the object's mass and shows how weight changes when the local field changes.
The straight line means that if the field halves, the weight halves too. The object's inertia does not change, only the force acting on it.
Rules
These are the compact results from the method above.
Weight vector
\[\vec W=m\vec g\]
Weight magnitude
\[W=mg\]
Spherical source
\[W=G\frac{Mm}{r^2}\]
Near Earth
\[W\approx mg,\qquad g\approx9.8\,\mathrm{N\,kg^{-1}}\]
Examples
Question
An
\[78\,\mathrm{kg}\]
explorer stands where the local field strength is \[3.7\,\mathrm{N\,kg^{-1}}\]
Find the explorer's weight.Answer
\[W=mg=78(3.7)=2.89\times10^2\,\mathrm{N}\]
directed with the field.Checks
- Weight is measured in newtons; mass is measured in kilograms.
- The same object has the same mass in every field but not the same weight.
- Weight points in the same direction as \(\\vec g\), toward the attracting body.
- A weaker field means smaller weight even when nothing about the object itself has changed.