AcademyForces and Newton's Laws
Academy
Weight Versus Mass
Level 1 - Physics topic page in Forces and Newton's Laws.
Principle
Mass measures inertia, while weight is the gravitational force produced by a local field.
Notation
\(m\)
mass
\(\mathrm{kg}\)
\(\vec W\)
weight force
\(\mathrm{N}\)
\(W\)
weight magnitude
\(\mathrm{N}\)
\(\vec g\)
gravitational field strength
\(\mathrm{m\,s^{-2}}\)
\(g\)
gravitational field magnitude
\(\mathrm{m\,s^{-2}}\)
\(N\)
normal reaction magnitude
\(\mathrm{N}\)
\(a_y\)
vertical acceleration with upward positive
\(\mathrm{m\,s^{-2}}\)
Method
Weight is a force model. Apparent weight is a contact-force reading found from the vertical force equation, so a scale does not always read the same value as \(mg\).
Local gravity model
\[\vec W=m\vec g\]
Weight magnitude
\[W=mg\]
Mass stays the same even when the local value of \(g\) changes.
Near Earth
\[g\approx9.8\,\mathrm{m\,s^{-2}}\]
Vertical force model
\[N-mg=ma_y\]
A scale reads the contact force \(N\), not the gravitational force directly.
Apparent weight
\[N=m(g+a_y)\]
This form uses upward as positive.
The free-body diagram shows why a scale reading can differ from weight during vertical acceleration.
The upward component equation compares the contact force with the gravitational force magnitude.
Rules
These are the compact results from the method above.
Weight vector
\[\vec{W}=m\vec{g}\]
Weight magnitude
\[W=mg\]
Near Earth
\[g\approx9.8\,\mathrm{m\,s^{-2}}\]
Scale equation
\[N-mg=ma_y\]
Examples
Question
Find the weight of a
\[72\,\mathrm{kg}\]
person near Earth's surface.Answer
\[W=mg=72(9.8)=706\,\mathrm{N}\]
downward.Checks
- Kilograms measure mass; newtons measure weight.
- Mass does not change when the local value of \(g\) changes.
- Near Earth, weight points downward toward Earth's centre.
- Downward acceleration can make the scale reading less than \(mg\).