Definition and Mathematical Expression

Definition of weight as a vector gravitational force and its scalar expression based on the local gravitational field.

Conceptual context

This page focuses on a specific, structural question: what it means to define weight as a vector and what changes when we work with its scalar magnitude. The key idea is not only to calculate a value, but to understand the conceptual architecture that connects $\vec{P}$, $P$, $\vec{g}$ and $g$, together with the parameters of the central gravitational model $G$, $M$ and $r$.

Unlike a purely operational approach, here the teaching priority is to distinguish between:

• The vector definition of the gravitational interaction.
• The scalar expression useful for quick calculation in a nearly uniform field.
• The extended central-field model that explains variations of gravity.

This distinction is the identity of the topic. It helps avoid recurring errors such as confusing mass with weight, or assuming that local gravity is identical at any altitude and on any planet.

🟢 Essential level

At the essential level it is worth fixing three ideas without algebraic saturation.

First idea: $m$ is an inertial property of the body and does not depend on location. Second idea: $\vec{P}$ is a gravitational force, so it depends on the gravitational environment. Third idea: the direction of weight always follows the direction of the local field, so when the reference geometry changes, its components change, but not its physical nature.

When a system of axes is chosen, weight can be represented by components. That decomposition creates no new forces: it only changes the mathematical language with which we describe the same interaction.

In everyday language: mass is "what the body is", weight is "how the celestial body pulls it". If the body or the distance changes, the weight changes; the mass does not.

A correct essential reading avoids two harmful shortcuts. First shortcut: thinking that every gravitational problem is solved with a single fixed value of gravity. Second shortcut: believing that changing notation changes nothing in the interpretation. In reality, choosing between a vector and a scalar description determines which questions can be answered rigorously. That is why this level insists on causal intuition before detailed calculation.

🔵 Formal level

Basic formulas

Before substituting values, identify which quantity each term represents, check that the units are compatible and predict the sign or trend of the result. This prior reading lets you use the relation as a physical model and quickly detect an incoherent substitution, direction or hypothesis.

The base vector relation of the topic is:

Vector definition of weight

The scalar form used in a nearly uniform field is:

Real student questions

Why is weight sometimes written as a vector and other times as a scalar? Because they answer different questions: the full direction versus the numerical intensity.

If the mass does not change, why does the weight change?

Because the gravitational field of the environment changes.

When do I have to use $G$ explicitly?

When $g$ cannot be assumed constant or when the problem requires a relation with $M$ and $r$.

Final synthesis

Defining and expressing weight correctly means choosing the right physical representation for the right question. $\vec{P}$, $P$, $\vec{g}$, $g$, $G$, $M$, $r$ and their components are not isolated pieces; they form a coherent conceptual system that allows calculating, interpreting and deciding with rigor.