Mass versus Weight

First, let's clarify the distinction between mass and weight. Mass refers to the amount of "stuff" contained in an object. If we liken objects to LEGO blocks, mass could represent the total number of blocks (albeit an imperfect analogy). In reality, objects are composed of protons, neutrons, and electrons—these subatomic particles contribute to an object's mass, measured in kilograms or grams.

When two mass-bearing objects are in proximity, they exert gravitational forces on each other. This attraction depends on the masses of both objects and the distance separating them. For example, a balloon resting on Earth experiences a gravitational pull from our planet, which we refer to as weight—this force is proportional to the object's mass (weight = mass × gravitational field). Weight is measured in Newtons.

If you transport that balloon to another planet, its weight may change based on the planet's size and mass, but its mass remains constant (the count of its components stays the same). As an extra tidbit, in imperial units, weight is expressed in pounds while mass is measured in slugs.

Balloon on a Balance

Now, imagine a scenario involving a balloon. When I place an uninflated balloon on the scale, it registers a reading of 2 grams. Now, consider this question: what happens to the scale's reading when I inflate the balloon and place it back on the scale?

Think about your prediction. Write it down—commit to it. Now, let’s test it.

After inflating the same balloon, the scale now reads 3 grams. Notice that I emphasize the scale reading rather than the actual mass; this distinction is deliberate. So, what accounts for the increase in the scale reading? Let’s create a force diagram for both the inflated and uninflated balloons resting on the scale.

In both instances, the balloon remains stationary, indicating that the net force is zero (according to Newton’s second law). Three forces act upon the balloon: the downward gravitational force (mg), the upward force exerted by the scale (denoted as N for normal force), and the buoyant force (F_B) arising from the interaction between the balloon and the air. The buoyant force’s magnitude depends on the air's density and the volume of the displaced air.

The density of air is approximately 1.2 kilograms per cubic meter, and V_b represents the balloon's volume. The inflated balloon has a significantly larger volume, resulting in a greater buoyancy force. However, both scenarios yield the same force equation (in the vertical direction, thus no vector considerations are needed).

Two critical observations arise: first, the scale's force (N) does not represent mass due to the buoyancy force's influence. For the uninflated balloon, its small volume makes the buoyancy force negligible, but this is not the case for the inflated balloon.

Second, the inflated balloon’s larger volume results in a greater buoyancy force. So why does the scale indicate a higher reading for the inflated balloon? The answer lies in the fact that air is trapped inside the balloon, requiring the scale to exert a greater force to balance the air's weight.

You might wonder why this doesn't simply balance out. The extra mass from the air in the balloon does not precisely negate the buoyancy force from its larger size. This discrepancy arises because the air within the balloon is at a higher pressure and, therefore, density than the ambient air. Additionally, when the balloon is inflated, its rubber stretches thinner, displacing slightly less air—though this is a minor effect.

Consequently, we see that the digital scale does not measure the mass (or even the weight) of the balloon. Instead, it provides a reading based on the force the scale exerts upward on the object, converted into grams.

Another Example

Let’s consider a simpler example: what if we position the balance upside down?

In this case, the scale displays a reading of 166 grams. That clearly doesn’t represent the tiny beaker placed on it. One could argue this reading reflects the balance's mass (and you would likely be correct). However, it again registers the force exerted by the beaker, which is equal to the force the beaker pushes upward against the balance (due to Newton's third law). Since the scale is at rest, the downward gravitational force and the upward force from the beaker must cancel out, resulting in a reading that reflects the balance's weight (in those peculiar units).

Balance on an Incline

Have you ever noticed that many digital scales come with a small leveling bubble? While the basic balance used in these examples lacks one, numerous models include leveling screws to ensure a flat operating surface.

Let’s see what occurs when we tilt the balance.

When tilted, the scale reads 84 grams, down from 103 grams when level. Should we gather some data? I will record the balance's readings at various tilt angles.

Interestingly, the results seem to resemble a cosine function. Let's plot the scale reading (which is not the mass) against the cosine of the angle. The outcome appears linear.

To understand the forces at play on a tilted balance, consider that the block remains at rest, meaning the net force is zero. Analyzing the components of the gravitational force perpendicular to the balance reveals that this component must equal N.

This leads to a conclusion: the scale should display a reading proportional to the cosine of the tilt angle, which aligns perfectly with our graph.

Note that F_f is the frictional force required to keep the block on the scale. To prevent it from sliding, I discreetly placed a small piece of tape under the block. However, it’s clear that the scale does not report the actual mass (or even the weight).

Let’s conduct one final experiment concerning weight.

Digital Scale in an Elevator

This one promises to be engaging. Take a digital scale with the same aluminum block and observe the reading as you ride in an elevator (either ascending or descending). If you lack a scale or elevator, here’s a demonstration.

Notice the reading starts at 102 grams, then briefly increases, returns to 102, and finally decreases. The question is: is the elevator moving up or down?

I won’t keep you in suspense. The elevator ascended one floor. To accomplish this, it had to initially accelerate upward. Here’s the force diagram during that acceleration.

Since both the block and the elevator accelerate upward, the force from the scale (N) must exceed the gravitational force. Consequently, the scale registers a higher value during this phase (which is not the actual mass of the block).

As the elevator approaches the second floor, it must decelerate, resulting in a downward acceleration. Since the gravitational force remains unchanged, the only way to account for this is for the scale to exert a smaller force (leading to a lower reading).

You might also consider invoking Einstein’s equivalence principle, which suggests that we cannot distinguish between gravitational force and an accelerated reference frame. This concept is applicable in creating artificial gravity in rotating spacecraft or explaining why astronauts feel weightless in space.

How Does a Balance Actually Work?

While I'm not entirely certain about this specific digital balance, many calculate "mass" (which we've established can be misleading) using a strain gauge. The fundamental principle is that the electrical resistance of a material can change when a force is applied.

Imagine a cylindrical metal wire. When a force is applied, the wire shortens slightly but expands in width (as volume is conserved).

When the wire shortens, its resistance decreases. Conversely, when it widens, its resistance also decreases. Thus, by measuring the changes in electrical resistance, one can deduce the applied force. This is the basic idea, though there are engineering considerations involved.

Since the strain gauge measures force, the digital balance effectively measures force. In all the examples provided, the balance force (N) varies, and that is what the reading reflects.

In scientific endeavors, it’s crucial to grasp the functioning of your instruments. They seldom deliver exactly what you desire, but with analysis, you can usually discern what you truly need.