KAHIBARO
Table of Contents
What Weight Means in Physics
Weight is the force with which gravity pulls on an object. If you hold a book in your hand, Earth pulls the book downward, and that downward pull is the book's weight.
Weight is a force, so it is a vector. That means it has both magnitude and direction. Near Earth's surface, the direction of weight is vertically downward, toward the center of Earth.
Many beginners confuse weight with mass. Mass tells how much matter an object has, while weight tells how strongly gravity pulls on that mass. A body can have the same mass in different places, but its weight can change if the gravitational field changes.
Weight is a force, not a measure of how much matter an object contains.
Mass is measured in kilograms, $\mathrm{kg}$.
Weight is measured in newtons, $\mathrm{N}$.
Formula for Weight
Near Earth's surface, the magnitude of the weight of an object is
$$
W = mg
$$
where $m$ is the mass of the object and $g$ is the gravitational field strength, often called the acceleration due to gravity.
On Earth, a common approximate value is
$$
g \approx 9.8\ \mathrm{m/s^2}
$$
So an object of mass $2.0\ \mathrm{kg}$ has weight
$$
W = (2.0\ \mathrm{kg})(9.8\ \mathrm{m/s^2}) = 19.6\ \mathrm{N}
$$
The weight force acts downward.
Near Earth, use
$$
W = mg
$$
and take the direction as downward.
Weight as a Vector
Because weight is a force, it can be written in vector form. If the upward direction is chosen as positive $y$, then weight points in the negative $y$ direction:
$$
\vec{W} = -mg\,\hat{j}
$$
This negative sign does not mean the weight is negative in a physical sense. It only shows that the force points downward in the chosen coordinate system.
Weight and Free-Body Diagrams
In a free-body diagram, weight is shown as an arrow starting at the object and pointing straight downward. This arrow is usually labeled $\vec{W}$ or $mg$.
This downward force is always present as long as the object has mass and is in a gravitational field.
Weight Versus Mass
The distinction between mass and weight is very important.
| Quantity | Meaning | Type | SI Unit |
|---|---|---|---|
| Mass | Amount of matter, measure of inertia | Scalar | $\mathrm{kg}$ |
| Weight | Gravitational force on the object | Vector | $\mathrm{N}$ |
A person with mass $70\ \mathrm{kg}$ does not have a weight of $70\ \mathrm{N}$. Their weight near Earth's surface is approximately
$$
W = (70)(9.8) = 686\ \mathrm{N}
$$
In everyday speech, people often say "I weigh 70 kilograms," but in physics that value is their mass, not their weight.
Do not say kilograms are units of weight in physics.
Kilograms measure mass.
Newtons measure weight.
Dependence on Location
Weight depends on the local value of $g$. Since $g$ is not exactly the same everywhere, weight can vary slightly from place to place.
For example, an object weighs less on the Moon than on Earth because the Moon's gravitational field is weaker. The mass stays the same, but the weight changes.
| Location | Approximate $g\ (\mathrm{m/s^2})$ | Weight of a $1.0\ \mathrm{kg}$ mass |
|---|---|---|
| Earth | $9.8$ | $9.8\ \mathrm{N}$ |
| Moon | $1.6$ | $1.6\ \mathrm{N}$ |
| Mars | $3.7$ | $3.7\ \mathrm{N}$ |
This is why astronauts feel much lighter on the Moon, even though their mass is unchanged.
Weight and Gravitational Force
Weight is a particular example of gravitational force. For many problems near Earth's surface, the simple formula $W = mg$ is enough. In more general situations, gravity depends on distance from a planet or star, but that belongs to the study of gravitation.
For now, it is enough to understand that weight is the gravitational force exerted on an object by Earth or another celestial body.
Common Example
Suppose a backpack has mass $5.0\ \mathrm{kg}$. Its weight near Earth's surface is
$$
W = mg = (5.0)(9.8) = 49\ \mathrm{N}
$$
So the Earth pulls downward on the backpack with a force of $49\ \mathrm{N}$.
If you lift the backpack and hold it still, its weight does not disappear. It still acts downward. Other forces may balance it, but the weight remains.
Key Idea
Weight is the gravitational force acting on an object. Near Earth's surface, it is found from $W = mg$, acts vertically downward, and is measured in newtons.
Important facts about weight:
$$
W = mg
$$
Weight is a force.
Weight points downward.
Weight depends on gravity.
Mass does not change when location changes, but weight can.