People watching their weight might claim that scales don't lie, but what they do tell a person is, at the very least, a misnomer. Weight, in physics terms, is actually a *force*: The force of gravity acting on a mass. The SI unit of force is Newtons (N). Mass, on the other hand is a measure of the amount of matter in an object. The SI unit of mass is the kilogram (kg).

So what the scale should really display to a person seeking their weight is a value in *Newtons*. For the discerning physics students who want to approximate this themselves; however, the following works: Just multiply the kilograms the scale gives by 10 (or the pounds by 4.5).

## What's the Difference Between Mass and Weight?

In a nutshell, the main difference between mass and weight is that mass is a *fundamental property* of an object and weight is not. Mass doesn't change no matter where an object is located until matter is added or subtracted from it. A 2,300 kg elephant is 2,300 kg on planet Earth, the moon and in the middle of space.

Weight, on the other hand, does depend on location since the gravitational force acting on the mass is different in different locations. A 2,300 kg elephant has a *weight* of approximately 23,000 N on Earth's surface, but only about one-sixth that weight on the moon and, if the elephant was deposited in deep space, far from the influence of any gravitational field, it would have no weight at all.

Another important distinction between mass and weight that follows from their definitions is that mass is a *scalar* value, as there is no direction associated with a value in kilograms, while weight is a force *vector.* An object's weight is always directed the same way as gravity pulls on it.

Mass technically is a quantitative measure of an object's inertia, or its resistance to motion. The more massive an object, the less it is affected by forces acting on it.

## Weight: The Force of Gravity

Like any force, weight can be calculated using the gravitational force equation:

Where *g* is the acceleration due to gravity near the Earth's surface: *g =* 9.8 m/s^{2}. Any object dropped anywhere on the planet falls toward the center of the Earth at an ever-increasing rate: 9.8 m/s faster each second than the previous second.

This formula explains why multiplying mass in kg by 10 (or in lbs by 4.5, to account for first converting to the SI unit of kg) gives a quick approximation of a person's "real" weight.

Elsewhere in the universe, the value of *g* is different, since acceleration due to gravity is a result of a large body's local gravitational field. On the tiny planet Mercury, for example, *g* is only 3.7 m/s^{2}. Because that's only about 38 percent of *g* on Earth, anything on Mercury weighs only about 38 percent of what it does on Earth.

## Apparent Weight

As a strict definition, an object's weight in the same gravitational field does not change. Whether a person is going up or down in an elevator, the same *g* is accelerating the same *m*, so *F _{grav}*, or weight, will be the same.

In reality, there are small differences in the value of *g* at different locations around a large body, such as at the North Pole versus the Equator on Earth, or in the interior versus on the surface of the Sun. But approximating a constant value for everywhere in a gravitational field is usually sufficient for physics students.

That said, observant elevator riders may have noticed they sometimes *feel* heavier or lighter than normal at different points in the ride. Their *apparent* *weights* are changing because their bodies have inertia, or they resist changes to their motion.

When an elevator begins rising, their bodies are still and resisting the upward motion, making them feel heavier for a moment until they adjust to moving. The reverse is true for a moment when the elevator starts descending. However, at no point did the person's *actual weight* change.

## Scales on an Accelerating Elevator

What about the reading of the scale for the same people going up and down the elevator? Here again, the scale might seem to lie, but this time not simply with a misnomer.

The scale operates by measuring the **net force** acting on it. When it is still on the bathroom floor, the entire net force on the scale is from the force of gravity pulling the body standing on the scale downward. But on an *accelerating elevator,* when the elevator starts speeding up or slowing down, the total acceleration of the mass on the scale is not just from *g* but also from the motion of the elevator.

If the elevator is accelerating upward in the opposite direction of *g*, the net acceleration will be slightly smaller than *g*, resulting in a slightly smaller net force (since *F _{net} = ma* and assuming the elevator's acceleration is smaller in magnitude than

*g*). The scale will therefore display a

**smaller number** than when it is still. Conversely, when accelerating downward, there is

*additional acceleration* in the direction of

*g,* resulting in a larger net force on the scale, and it will display a

**larger number**.

Note that this is *only true when the elevator is accelerating*. At a constant velocity up or down (which most passengers might hope for!), the net acceleration and thus the net force does not differ from the scale that is not moving on the bathroom floor.

## Scales on an Incline

Another easy way to instantly "lose weight" is to put a scale on an incline rather than flat on the floor. Drawing a free-body diagram of the forces on the scale, and understanding how the scale works, reveals why this is true.

Again, the scale works by registering the force of gravity acting on it downward into the scale. The force of gravity is always directed toward the center of the Earth. When the scale is flat on the bathroom floor, this is directly downward at 90 degrees.

When the scale is tilted though, for instance sitting on a ramp at 20 degrees, the force of gravity is *no longer perpendicular to the scale*. Resolving the force of gravity into its components reveals that *the perpendicular component,* the one that goes straight into the scale and thus serves as the source of the scale's reading, is **smaller than the total force of gravity**. Thus, the scale displays a **smaller number** when inclined than when flat on the floor.

## Why Knowing the Difference in Mass vs. Weight Matters

Mass and weight **are not** interchangeable in physics! Many equations and concepts depend on an object's mass or on multiple objects' masses. Weight is only a useful concept in Newtonian physics situations, like analyzing forces in the situations described here.

References

About the Author

Amy Dusto is a high school science teacher and a freelance writer. She holds a Bachelor of Arts in Natural Sciences area and a Master of Arts in Science Writing from Johns Hopkins University. She has contributed to Discovery.com, Climate.gov, Science News and Symmetry Magazine, among other outlets.