When you weigh yourself, you stand on a spring scale and compress a spring. The reading of a spring scale, such as the one shown in FIGURE 6.9, is $F_{\text{Sp}}$, the magnitude of the upward force the spring is exerting.

FIGURE 6.9 A spring scale measures weight.

With that in mind, let’s define the weight of an object to be the reading $F_{\text{Sp}}$ of a calibrated spring scale when the scale is the object’s only support and the object is at rest relative to the scale. That is, weight is a measurement, the result of “weighing” an object. Because $F_{\text{Sp}}$ is a force, weight is measured in newtons.

If the object and scale in FIGURE 6.9 are stationary, then the object being weighed is in equilibrium. $\vec{F}{}_{\text{net}}=\vec{0}$ only if the upward spring force exactly balances the downward gravitational force of magnitude $mg:$

Because we defined weight as the reading $F_{\text{Sp}}$ of a spring scale, the weight of a stationary object is

Note that the scale does not “know” the weight of the object. All it can do is to measure how much its spring is compressed. On earth, a student with a mass of 70 kg has weight $w=(70\text{kg})(9.80\text{m}\text{/}{\text{s}}^2)=686\text{N}$ because he compresses a spring until the spring pushes upward with 686 N. On a different planet, with a different value for $g$, the compression of the spring would be different and the student’s weight would be different.

Surprisingly, you cannot directly feel or sense gravity. Your sensation—how heavy you feel—is due to contact forces pressing against you, forces that touch you and activate nerve endings in your skin. As you read this, your sensation of weight is due to the normal force exerted on you by the chair in which you are sitting. When you stand, you feel the contact force of the floor pushing against your feet.

But recall the sensations you feel while accelerating. You feel “heavy” when an elevator suddenly accelerates upward, but this sensation vanishes as soon as the elevator reaches a steady speed. Your stomach seems to rise a little and you feel lighter than normal as the upward-moving elevator brakes to a halt or a roller coaster goes over the top. Has your weight actually changed?

To answer this question, FIGURE 6.10 shows a man weighing himself on a spring scale in an accelerating elevator. The only forces acting on the man are the upward spring force of the scale and the downward gravitational force. This seems to be the same situation as Figure 6.9, but there’s one big difference: The man is accelerating, hence there must be a net force on the man in the direction of $\vec{a}$.

FIGURE 6.10 A man weighing himself in an accelerating elevator.

For the net force $\vec{F}{}_{\text{net}}$ to point upward, the magnitude of the spring force must be greater than the magnitude of the gravitational force. That is, $F_{\text{Sp}}>mg$. Looking at the free-body diagram in FIGURE 6.10, we see that the y-component of Newton’s second law is

where m is the man’s mass.

We defined weight as the reading $F_{\text{Sp}}$ of a calibrated spring scale when the object is at rest relative to the scale. That is the case here as the scale and man accelerate upward together. Thus the man’s weight as he accelerates vertically is

If an object is either at rest or moving with constant velocity, then $a_y=0$ and $w=mg$. That is, the weight of a stationary object is the magnitude of the (effective) gravitational force acting on it. But its weight differs if it has a vertical acceleration.

You do weigh more when accelerating upward $(a_y>0)$ because the reading of a scale—a weighing—increases. Similarly, your weight is less when the acceleration vector $\vec{a}$ points downward $(a_y<0)$ because the scale reading goes down. Weight, as we’ve defined it, corresponds to your sensation of heaviness or lightness.*

We found Equation 6.10 by considering a person in an accelerating elevator, but it applies to any object with a vertical acceleration. Further, an object doesn’t really have to be on a scale to have a weight; an object’s weight is the magnitude of the contact force supporting it. It makes no difference whether this is the spring force of the scale or simply the normal force of the floor.

* Surprisingly, there is no universally agreed-upon definition of weight. Some textbooks define weight as the gravitational force on an object, $\vec{w}=(mg\text{,}\text{down)}$. In that case, the scale reading of an accelerating object, and your sensation of weight, is often called apparent weight. This textbook prefers the definition of weight as being what a scale reads, the result of a weighing measurement.