# Flat-earth gravity

#### Prerequisites

One of the forces we listed in going through our list of what interactions could change an object's velocity was **weight**. Every object seems to be pulled straight down, and if we hold it out and release it, it will fall to the ground. Let's try to make sense of what this force is and how it behaves by doing some observations (phenomenology). Although we have lots of experience with gravity, reconciling our experience with the framework of Newton's laws is a bit tricky. Our brain tends to create useful shortcuts that quickly give us the right answer in particular situations, but that don't generalize well and may be inconsistent with each other. Look at how many dangerous bends there are in the text below!

If we drop a variety of objects, we observe that they all fall to the ground, but some fall more slowly than others. A crumpled up piece of paper will fall more slowly than a lead brick. But we know that in addition to the object being pulled down — its weight — it's touching something: the air. And the air can exert a resistive force on the object (viscosity or drag). In order to try to learn just about gravity and not about a mixture of gravity and drag, let's focus on objects where it seems that the effect of air resistance is much less than the effect of gravity. And once we have a good idea about how gravity works, we'll go back and explore objects falling with air resistance.

For this page, then, we'll look at objects for which air resistance seems to have not much of an effect. This means we'll use dense objects and we won't drop them very far — no more than a couple of meters. Imagine a series of spheres about 2 inches in diameter having masses ranging from about 1/4 of a kg to a couple of kgs. And we'll drop them only at most a few meters. This means we can ignore the 1/*r*^{2} fall off of gravity, since the *r* is our distance to the center of the earth — about 2/π x 10^{7} meters — and we won't be changing that distance by a significant amount.

Since air resistance depends on velocity, this also means we won't look at objects going very fast. We'll also restrict our observations to a lab room -- not changing our position or height very much. Since this turns out to ignore effects of our position relative to and our distance from the center of the earth, we'll call this approximation *flat-earth gravity.*

[This is a nice example of "choosing a channel on cat TV". By restricting our considerations to objects for which air resistance is not important, we can get a good idea about the gravitational force. We can then expand our channel and add in air resistance. Finally, we can explore what happens as we get far from the earth and learn about Newton's Universal Law of Gravitation. If we worried about all this at once, it would be way too confusing.]

## How does the weight depend on position and time?

We observe from our everyday observations that all of the objects we are looking at have weight. We can feel them pulling down on our hands when we hold them. Since the object is exerting a normal force down on our hand, our hand must be exerting a normal force up on the object, by Newton's 3^{rd} law. Since the object isn't accelerating, the forces on it must be balanced. This means that the object's weight equals the upward normal force of our hand (by Newton's 2^{nd} law). So all three forces must be equal in magnitude. This tells us that the force we feel the object exerting on us is equal to the object's weight.

Careful! We often use the shortcut saying that we feel the object's weight with our hand. But since significant weight is only exerted by the earth, what we are actually feeling is the normal force the object exerts, which is equal to its weight. This is important because the chain of reasoning only holds in this static situation.

We can therefore measure this force using a spring (scale) instead of our hand, or we can drop it and measure its acceleration. What we will find if we stay in our room is:

The gravitational force on an object always points straight down and is the same no matter where in the room we put it -- or when. (Better: In flat-earth gravity approximation, the gravitational force on an object is straight down and independent of position of the object or when it is measured.)

## How does the weight depend on the object?

Now the interesting thing to determine how the weight depends on the properties of the object. One way to test this is to drop different objects and see how they accelerate differently. Careful! We are still in the flat-earth-negligible-air-resistance channel of cat TV so we have to restrict our experiments to these situations.

If we do the experiment of dropping a heavy and a light ball, we find the result that they hit the ground at the same time. If we watch a videotape it shows that they are accelerating together. This seems a little strange. What does it imply? Let's analyze it in our theoretical framework of Newton's laws.

To use Newton's laws on a falling body, we have to first do a free-body diagram and identify all the forces on the object. Since we have specifically said we have chosen objects for which we think air drag can be neglected, that force shouldn't matter much. All we have is the weight, since nothing else is touching it. N2 then becomes:

$$a_A = F^{net}_A /m_A = W_{E \rightarrow A} / m_A$$

The acceleration of the object equals the net force divided by the mass of the object. In this case the only force is the weight, so we get that the weight divided by the mass is the acceleration, which is now seen to be independent of the object by experimental observation.

So let's write $W_{E \rightarrow A} / m_A$ = constant independent of the object. We'll call it $g$. This gives:

$$W_{E \rightarrow A} = m_A g$$

The weight of the object is proportional to its mass. The constant g is called the *gravitational field*, and clearly has units of Newtons/kg. Measurements give

$g = 9.81$ N/kg

I like to write $g$ this way since it reminds us that it's about a force. The weight is a force. And it also reminds us that we have to multiply g by a mass in order to get to a force. A disturbing fact can be noticed right away from the result $a = W/m = g$, namely, that $g$ also has units of acceleration — meters/second^{2}. If you use the fact that a Newton = 1 kg-m/s^{2}, you will see this come out. But conceptually, $g$ is always about force. It only turns into an acceleration in the very special case of free-fall — when there are no other forces acting on the object. This can be very confusing! For example, an object sitting still on a table has a weight — and therefore a non-zero value of $g$ — but it has an acceleration = 0. It's MUCH better to use units of N/kg for g rather than the traditional "m/s^{2}".

## Watching out for one-step thinking

Here's a great YouTube video on the subject done at the University of Sydney in Australia. Most of the people asked about which would fall first, the heavy or the light ball, made the mistake of assuming that it would be the heavy ball. Do you think they are suffering from one-step thinking? Watch the video before reading my analysis below.

This is one of the most "dangerous bends" in Newtonian physics and I suspect it comes from one step thinking: bigger force means bigger acceleration. Well that's true if your thinking about a single object. Put more net force on an object and you will get more acceleration. But that's NOT what we're doing when we compare two objects.

Newton's second law reminds us the fundamental conceptual idea that the net force on an object is *shared* over all parts of that object. This is the idea of *mass. *Think about kicking a cannonball and a soccer ball. Which would go faster if you delivered each the same force with your kick? Everybody knows that the soccer ball would accelerate more and go faster! But somehow, when it's turned sideways and dropped, people (as in the video) assume a heavier object with an equal force would accelerate just as fast. Newton's 2nd law allows us to keep our two intuitions — about the effect of more force on acceleration and the effect of more mass on acceleration — and to reconcile them.

Joe Redish 10/2/11

#### Follow-ons

Last Modified: July 12, 2019