Weight force

When you step on a scale, the number you see represents the gravitational force that Earth exerts on you.

This value, commonly known as "weight," has a precise nature: it is a force, and in the International System (SI), it is measured in newtons (N).

In essence, the weight of an object is the force that Earth uses to pull it toward its center.

For instance, if you place an object on a scale and it reads 5 N, it means Earth is exerting a force of 5 N on that object. If you add another object of identical mass, doubling the total mass, the scale will show double the weight: 10 N.

Weight and mass are directly proportional, and the relationship is expressed by the formula:

$$ P = mg $$

Where:

P is the weight, measured in newtons (N)
m is the mass, measured in kilograms (kg)
g is the gravitational acceleration, which averages around 9.81 N/kg on Earth’s surface.

This means that an object with a mass of 1 kg weighs approximately 9.81 N.

The value of $ g = 9.81 \, \text{N/kg} $ can vary slightly depending on where you are on Earth. For example, $ g $ is slightly higher at the poles because Earth is slightly flattened there, reducing the distance to its center. At the equator, $ g $ is lower. Similarly, at higher altitudes, $ g $ decreases as the distance from Earth’s center increases. For instance, an object weighing 100 N at sea level will weigh slightly less at the top of a mountain.

As a force, weight is a vector quantity $ \vec{P} $, meaning it has:

Magnitude: the strength of the force (calculated as $ mg $)
Direction: perpendicular to Earth’s surface
Sense: always pointing toward Earth’s center.

This detailed representation helps us better understand how weight interacts with the environment and other forces.

example of a vase on a table

Like all vectors, the weight force can be broken into components.

For example, when an object is placed on a slope, it is subject to the weight force $ \vec{P} $, which pulls it downward, toward Earth’s center.

example of weight force decomposition

This weight force $ \vec{P} $ can be split into two components that affect the object’s motion along the slope and the pressure it exerts on the inclined plane.

One component $ \vec{P_y} $ is parallel to the slope’s surface, while the other $ \vec{P_x} $ is perpendicular to it.

Parallel component ($ \vec{P_y} $)
The vector $ \vec{P_y} $ runs parallel to the slope’s surface, pointing downward along the incline. This component drives the object’s motion along the plane. Its magnitude is calculated as: $$ P_y = P \sin\theta $$ where $ \theta $ is the slope’s angle of inclination relative to the horizontal.
Perpendicular component ($ \vec{P_x} $)
The vector $ \vec{P_x} $ is perpendicular to the slope’s surface, directed inward. This component represents the force pressing the object against the slope. Its magnitude is given by: $$ P_x = P \cos\theta $$

This decomposition is essential for determining both the object’s motion and the normal force exerted by the slope.

In these formulas, $ P $ represents the total weight, calculated as $ P = m \cdot g $, where $ m $ is the object’s mass and $ g $ is the gravitational acceleration.

For example, imagine an object with a mass of $ m = 10 \, \text{kg} $ on a slope inclined at $ \theta = 30^\circ $. The total weight is: $$ P = m \cdot g = 10 \, \text{kg} \cdot 9.81 \, \text{N/kg} = 98.1 \, \text{N} $$ The parallel component is: $$ P_y = P \sin\theta = 98.1 \, \text{N} \cdot \sin 30^\circ = 98.1 \, \text{N} \cdot 0.5 = 49.05 \, \text{N} $$ The perpendicular component is: $$ P_x = P \cos\theta = 98.1 \, \text{N} \cdot \cos 30^\circ = 98.1 \, \text{N} \cdot 0.866 = 84.88 \, \text{N} $$

In the same way, the gravitational force acting on a person climbing a mountain can be divided into two components.

The first component, $ \vec{P_y} $, is parallel to the surface of the rock face, while the second component, $ \vec{P_x} $, is perpendicular to it.

example of gravitational force

Difference Between Weight and Mass

In everyday conversations, "weight" and "mass" are often used interchangeably, but in physics, they have distinct meanings.

Understanding this difference is key to grasping the concepts of gravity and force.

Mass
Mass is a fundamental property that measures the amount of matter in an object. It is expressed in kilograms (kg) in the International System and remains constant regardless of location—on Earth, the Moon, or in deep space. Mass also determines an object’s inertia, its resistance to changes in motion.
For example, an object with a mass of 100 kg has the same mass on Earth as it does on the Moon.
Weight
Weight, on the other hand, depends on the gravitational force acting on an object’s mass. For instance, an object with a mass of 100 kg will have different weights on Earth and the Moon due to the difference in gravitational acceleration. On Earth, the gravitational constant is $ g = 9.81 \, \text{N/kg} $, so the object’s weight is: $$ P_{\text{Earth}} = m \cdot g = 100 \, \text{kg} \cdot 9.81 \, \text{N/kg} = 981 \, \text{N} $$ On the Moon, where $ g = 1.62 \, \text{N/kg} $, the weight is significantly lower: $$ P_{\text{Moon}} = m \cdot g = 100 \, \text{kg} \cdot 1.62 \, \text{N/kg} = 162 \, \text{N} $$ Thus, while the object’s mass remains constant (100 kg), its weight varies based on the gravitational acceleration of the celestial body.