A Formula with Conditions

In the college laboratory stands a cart with a load. Timur and Dana are preparing an experiment, and meanwhile going over a worked example. The cart’s mass with the load is 19.0 kg. A string pulls it horizontally, and it moves in a straight line with an acceleration of 1.5 m/s². What force must be applied?

“Mass times acceleration,” says Timur. “Nineteen by one and a half. Twenty-eight and a half newtons.”

“And if the dynamometer on the string reads thirty?”

Timur checks the multiplication. It still comes out the same.

“Then the dynamometer is lying.”

His arithmetic is right. But before blaming the instrument, look at which force Timur computed and which Dana is about to measure.

What hides behind the letter F

In Newton’s second law, force is the resultant of all external forces. They add up with directions allowed for. For our problem pick the system: the cart with the load. The string pulls it forward, and resistance forces point backward. Gravity and the support force cancel each other in this example: there is no vertical acceleration.

We care about motion along the horizontal track. In short: F = ma, where F is the resultant along the track, m is the mass of the cart with the load, and a is its acceleration. Substitution gives:

19.0 kg × 1.5 m/s² = 28.5 N.

The dynamometer on the string measures tension. To get the resultant from it, the remaining forces must be allowed for. If the string pulls with 30 N, say, and total resistance is 1.5 N, the resultant is 28.5 N. Both numbers agree with the calculation.

The 1.5 N of resistance we have just assumed for the example. To explain a real instrument’s reading, it will have to be determined. Fitting the gap between two numbers does not yet establish the cause of the disagreement.

Timur answered the question about the resultant. The question about the string’s tension stays open: with no information on resistance, one product of mass and acceleration is not enough for it.

Conditions beside the formula

Beyond the meaning of the letter F, two more clarifications are needed.

What happens to the mass? We treat the cart and its fastened load as one system of constant mass. Through the experiment it loses nothing and picks nothing up on the way. The familiar writing of the second law fits such a description.

But if sand pours out of the cart, or a rocket throws off gases, motion with departing matter must be taken apart, allowing for what crosses the system’s borders and how it moves. Simply plugging a changing mass into the familiar formula, without asking what crosses the system’s borders and how it moves, will not do.

Measured against what is the acceleration? In our example, against the laboratory. For this experiment we count its frame of reference as inertial: a body with zero total external force would keep its velocity there in size and direction. The earthbound laboratory meets this condition only approximately, but for an ordinary cart experiment the approximation is enough.

An accelerating railway car shows why this matters. Imagine the cart first resting in the car, able to roll with practically no resistance. The car starts accelerating forward. The cart lags behind it, and a passenger sees it move backward. That takes no force pulling it toward the rear wall: it is the car’s motion that changes.

The passenger may describe what happens against the car. But then the equation of motion needs an inertia force, tied to the acceleration of the chosen frame. Another way is to reckon against a frame that can be counted as inertial. What matters is stating which way we picked.

The formula takes one line. The clarifications run longer, because they tie its letters to a particular cart, its forces, and a way of watching motion.

When numbers disagree

Now change a condition. Let 30 N be named the resultant itself, and take the 19.0 kg mass and 1.5 m/s² acceleration, for a start, as exact stated quantities.

These three values do not satisfy F = ma: the product is 28.5 N. We have found an inconsistency. Keeping the formula and its conditions of use, all three numbers cannot be accepted at once.

But the computation does not say what exactly to fix. Perhaps the mass is misstated. Perhaps the wrong acceleration got copied. Perhaps tension was again called the resultant. We must go back to where the data came from and what it means.

With real measurements one more question appears. Mass, acceleration and force are known with uncertainty. So the comparison must weigh not only two numbers — 28.5 and 30 — but the uncertainties of the independently measured force and of the computed result. The latter depends both on measuring mass and on measuring acceleration. Under a shared measurement method the ties between the input data have to be allowed for too.

Until that information is in, we cannot say whether the gap matters. First make sure the same quantity is compared, then estimate how exact the comparison is. We will come back to this in the experiment chapter.

Newton’s second law cannot be declared refuted over two differing readings. But neither may the experiment’s own description be placed beyond checking in advance. A gap is a reason to check the data, the counted forces, and the accepted approximations.

Units join the calculation too

Even before the experiment we can check whether the formula yields a quantity of the wanted kind. In our product kilograms multiply by metres per second squared. That is exactly what a newton is:

1 N = 1 kg·m/s².

If a computed force comes out written in seconds, something is wrong already: in the formula, the substitution, or the answer’s entry. No dynamometer is needed for that verdict.

Take an even simpler case. At a steady 2 m/s for 3 s a body covers 6 m. Speed times time gives length. The answer “6 seconds” fails on dimension. But “9 metres” is wrong too, though its dimension fits. Checking units sifts out some errors; by itself it proves neither the number right nor the formula applicable.

In a school problem the 28.5 N result is usually rounded to 29 N, keeping two significant figures — as many as the 1.5 acceleration entry holds. That is the rounding rule accepted for such a calculation. In a measurement report the rounding place is matched to the result’s evaluated uncertainty.

And in both cases the unit stays beside the number. “29 N” names a force. A lone “29” does not yet say what we found.

What exactly the calculation grounded

Timur adds the word “resultant” to his answer.

“So we leave the dynamometer alone for now?”

“First let us find out what to compare its reading against,” says Dana.

Now the answer can be traced from start to finish. We picked the cart with the load, stated constant mass and a frame of reference, named which forces we add. Under these conditions the second law gave 28.5 N. We checked the multiplication and the units.

Such a calculation shows what follows from accepted premises. Seeing how well it describes the real cart takes measurements and a check of the assumptions. And grounds to trust the physical law itself come from experiment: correct multiplication cannot prove a law of nature.

In the next chapter we move to a triangle on paper. There too are a formula and conditions, but another way of grounding appears — mathematical proof. And no ruler, however good, replaces it.