III) Mass, Weight and Gravity at Earth's Surface: Key Ideas

People sometimes use the words mass, weight and gravity interchangeably. Often this is because people don't really understand the subtle differences in the meaning of these words. It is true that there are some similarities, but it is their differences that are most important. Here we will look at the importance of these concepts.

A) Mass vs. Weight

To understand gravity, we must understand the difference between mass and weight. To most people this seems like a matter of semantics, since we use the terms almost interchangeably. One reason for this confusion is that people are somewhat familiar with the pound (unit of weight in the English system) and the kilogram (unit of mass in the metric system). They are, however, largely unfamiliar with the slug (unit of mass in the English system) and the Newton (unit of weight/force in the metric system). These terms are further confused by U.S. students in particular because the movement to the metric system in the United States has never been very successful.

To a scientist, however, the difference between weight and mass is of crucial importance. Loosely speaking, mass is the amount of material in an object. A large object with low density (say a boulder made of Styrofoam) might have the same amount of mass as a small object of high density (a small lump of lead). The mass measures how hard it is to start an object moving or to slow it down again. Pushing a car is hard work (even on a flat road) because it is very massive.

So what is weight? Weight is the force of attraction between an object and whatever astronomical body it is on or near. Notice that the mass is simply a property of the object itself, but the weight is a property of the object and its location. The important thing is that moving a car in zero-gravity would be just as much work as it is on Earth, because the car's mass isn't any different, even though its weight would be zero. It would take a great deal of work to get the car moving in the first place, due to its mass. Astronauts routinely complain that it is awfully hard work to move around or work on the satellites they have to deal with. Even though they and their tools are weightless, they are not without mass; therefore, they still require effort to move.

However, "weightlessness" is not the absence of the force due to gravity. It occurs anytime one is falling! In orbit, one is falling continuously (but missing the Earth!), and so one doesn't feel the gravity. But it is there. Otherwise one would fly off into space. Gravity does not end outside of the Earth's atmosphere.

B) Mass and Gravity

Near the surface of the Earth, gravity as a force has a very simple expression:

Where "little g" is a constant equal to about 9.8 m/s². Note that the units are those of acceleration. What are the consequences of this force law?

Most apparent is that gravity, as noted above, is responsible for weight. All objects on Earth experience a force pulling in a direction we perceive as down. But force is any change in motion. Why don't we start moving down? We would, but for most objects on Earth the force of gravity is balanced by some upward force: you don't sink through the floor because the floor is pushing up against your feet!

Notice that the force of gravity (weight) is proportional to the mass of the object. This is why we frequently confuse mass and weight in our speech and even in our minds. If the force of gravity were dependent on some other property (for example the color) then mass and weight would not be confused.

Your weight is found by stepping on a simple bathroom scale and standing still. Since your weight is proportional to your mass (and on Earth it will never change for a given mass), the scale can be calibrated to tell you your mass (typically in kg) and weight (typically in lb) at the same time. You can see how most people will ignore the differences and just assume they are the same. Gravity begins to be more interesting when one starts to look at how objects move under its influence.

C) Gravity Predicts the Motion of Objects on Earth

Gravitational forces allow us to understand and predict the motion of most objects we encounter in nature. The simplest case is that of dropping an object, say a water glass. When the balancing force exerted by your hand is removed, the net force on the water glass is given solely by gravity:

Applying Newton's second law (Force = mass * acceleration) to find the motion resulting, we can substitute ma for F:

We can see that the acceleration of the water glass is equal in magnitude to g. Notice that this is completely independent of the mass of the object! This of course is just Galileo's discovery that objects of different masses fall at the same rate (ignoring air resistance). But acceleration is just the rate of change of velocity so we have

We can then either apply a geometric argument (such as various medieval philosophers did) or use an application of calculus by taking the anti-derivative (integrate) with respect to dt of both sides. Either way we obtain,

which is again what Galileo obtained (though with much more effort since he didn't really have a proper understanding of acceleration or velocity). Since the vertical motion (vy) of an object is independent of the horizontal motion (vx), trajectories under gravity are parabolic:

The classic demonstration of this independence is the "monkey and the hunter" problem. A monkey is aiming at a hunter she has chased up a tree. At the instant the gun goes off the hunter hears the gun, sees the flash, and drops, hoping to avoid being hit. What happens? Does the hunter escape injury or not? The trick of course is realizing that all objects accelerate at the same rate at the Earth's surface regardless of their velocities. Thus the bullet and the hunter would both fall at the same acceleration for the same time and drop the same distance. If the monkey's aim were true in the absence of gravity, then the unfortunate hunter would be hit even with gravity.

D) Gravitational Force and Energy

Force and momentum are two of the key concepts of physics. Equally important, however, is energy. How are energy and gravity tied together?

Lifting an object requires work. Let us consider this in detail. Imagine that you have lifted an object of mass m, to a height of meters above the floor. You now let it drop. As discussed above, it will start falling with an acceleration g. How fast will it be going when it reaches the floor? The amount of time it takes can be obtained,

at which point it will be moving at a speed

We can calculate the kinetic energy of the object using the usual formula

Where did this energy come from? It must have come from the work it took to lift the object in the first place! The work exerted to lift an object doesn't disappear; it simply turns into potential energy, ready to be converted back into kinetic energy at a moment's notice. On the other hand, imagine that the falling object is actually a superball. How high will it bounce? If it is a really good superball then it will bounce all the way back up to the original height (minus a bit for friction)! So the Kinetic energy is converted back into potential energy (U)

and the cycle is ready to repeat. Note that the potential energy can also be thought of as the force (mg) times the distance ( ), as you might expect, since those are the units of energy.

The total energy of an object at a height h is

which depends only on the original height! Conversely, if one has the total energy of an object, one can predict the height to which it will rise.

E) Gravity Predicts that All Objects Near the Earth's Surface Fall with the Same Acceleration

When Galileo performed his groundbreaking experiments with falling objects, he noticed a very odd fact: all objects fall at the same rate regardless of the mass. This is very strange... If I hold a heavy object in my hand, that object presses down with much greater force than a light object would. So, why doesn't a heavier object fall faster?

In a previous section, we saw that the mass of the object in the expression for gravitational force cancels with the mass in the equations of motion. Heavier objects experience greater gravitational force but are also harder to push. Hence, the kinematics of falling bodies are independent of mass.

This explanation is very clever but it leaves something unexplained: why should both laws involve the very same quantity? Motion and gravity seem to be two very different things. As mentioned in the Historical Perspectives section, scientists call the mass in the Law of Gravitation "gravitational" mass and the mass involved in motion "inertial" mass. The experiments performed by Eotvos and since confirmed repeated, show that gravitational mass is equal to inertial mass to an accuracy of at least a few parts in a hundred million! This is true for all types of objects regardless of what they are made. So we have solved one mystery and understood the Newtonian connection between gravity and motion at the cost of another mystery: why are the gravitational and inertial masses so very precisely the same? For years no one knew what to make of this until Einstein came along with General Relativity.

In General Relativity, gravity is the curvature of space due to the presence of matter. The path of an object is bent in a gravitational field not because of a force acting on it, but because the space it is traveling in is itself bent. But all objects travel through the same space, so they all move in the same way. The acceleration of an object is due to a property of the space, not of the object. Thus, it must be the same for all objects, regardless of mass or other characteristics.

In this chapter, we have discussed gravity near the Earth's surface. Gravity, however, is universal. Every object exerts a gravitational attraction on every other object in the universe. And this means everything...every grain of sand, every drop of water, every atom of hydrogen in interstellar space, even the smallest of subatomic particles... everything. It doesn't matter what else is true about an object. It doesn't matter if it is pink with purple polka dots, if it looks vaguely like Elvis, or if it is made of Swiss cheese. The only thing that's important is the mass. No special properties of objects are invoked. Gravitation is truly Universal!