The Physics
Opus in profectus


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Why does this page exist?

This is not a page about some fundamental principle of physics. It's a page about solving a particular (and common) kind of problem in mechanics.

Informally, statics is the study of forces without motion. More formally, statics is the branch of mechanics that deals with forces in the absence of changes in motion. In contrast, dynamics is the study of forces and motion; or more formally, the branch of mechanics that deals with the effect that forces have on the motion of objects. Statics implies stasis. Dynamics implies change. The change that matters is acceleration.

Statics seems to imply being stationary, but this is not necessarily the case. Acceleration is what matters. Sitting still for an extended period of time is one way to not accelerate. Moving at a constant speed in a straight line for an extend period of time is another way. There is no physical distinction between being at rest and moving with a constant velocity. The Earth is faithfully following the Sun through the Milky Way at an inconceivable speed (250,000 m/s), but because our acceleration is nearly zero (~10−10 m/s2) this motion is essentially undetectable. When we are at rest on the moving earth, we feel as if we are at rest with respect to the entire universe.

The purpose of this section of this book is to serve as a repository for problems in statics. The acceleration in every problem will be zero in all directions. That is only true for this section. The idea is to see what it's like to solve such problems so you can recognize them when they pop up later.


equal balance

When no forces act on an object, it does not accelerate. This is a consequence of Newton's first law of motion. "An object at rest tends to remain at rest and an object in motion tends to continue moving with constant velocity." This seems to imply that when one or more forces do act on an object, it must accelerate. This is not quite true, however, because it overlooks the last bit of the law — "unless compelled by a net external force to act otherwise".

A single force will accelerate an object. This is easy to test. Drop something. Watch it accelerate. Well, maybe it's not so easy to test, since we seem to drop nearly everything in air near the surface of the Earth. That means drag is present as a second force. True, but we could think about the limiting case. When drag is large, free fall acceleration can be hard to perceive. When drag is small, acceleration is obvious. When it's smaller still, it's only more obvious. Extrapolating to the point where weight is the only force acting on an object does not make acceleration go away. It only makes it more apparent.

So what about the case of two forces, or three, or more? This again is easy to test. Sit in a chair, stand on the floor, or lie in bed. There are two forces acting on you (weight pulling down and the normal force pushing up) and no acceleration. There are several ways to describe this situation.

Let's deal with these in order.

This is my personal favorite (which is why I put it first) because it relates back to the statement of the first law (which is also why I put it first). The word "net" in the phrase "net force" means total, combined, or overall. It's what you get when all things are considered. The word net is related to the word neat. Finding a net value is something like cleaning up a mathematical mess (or at least reducing a mess).

A person sitting in a chair is acted upon by two forces whose sum is zero. They sum up to zero because one force is positive and one is negative. Forces are vectors — they have direction as well as magnitude. The simplest way to describe direction is with a mathematical sign. The net force on an object is the vector sum of all the separate forces acting on it. This can be written as F or Fnet or in other similar ways.

When a person sits in a chair, their weight is balanced by the normal force from the chair. Weight and normal in this case are said to be balanced forces because one is equal and opposite the other. This leads to confusion sometimes because Newton's third law of motion says every action has an equal and opposite reaction. All action-reaction pairs of forces are equal because that's the way the universe works. Weight and normal are not an action-reaction pair, however. If these two forces are equal on the person, it's only because the person is not accelerating.

The gravitational pull of the Earth on the person and the person on the Earth is an action-reaction pair. The two forces are acting on two different objects (the person and the Earth). This does not make weight a balanced force. The normal force of chair on the person and the person on the chair is another action-reaction pair. Again we have two forces acting on two different objects (the person and the chair). This does not make the normal force a balanced force either. The pull of gravity on the person and the normal force on the person do act on the same object (the person) and they combine together to result in an object is not accelerating. These forces are balanced.

A person sitting in a chair is in equilibrium. Now we have a legitimate word to chew on. An object that is not accelerating can also be said to be in equilibrium. The first part equi- should be easy to identify as the suffix form of the English word equal, which is from the Latin word aequalis. The second part -librium comes from the Latin word libra, which refers to a balance — a device for determining weight by comparison that uses two pans. Think of the scales of justice when you see the word equilibrium.

A double pan balance

As a fun aside, the Latin word libra is also the origin of the ornate capital L symbol for the current British pound (£) and the former Italian lira (₤). It's also the origin of the unusual symbol for the British-American pound of weight (lb or ℔). The version with the horizontal strikethrough evolved into the pound symbol that is now only seen on telephone keypads (#). With bad handwriting, a weight like 24 lb might be read as 24 16. Writing it as 24 ℔ added some clarity, but it also added an extra pen stroke. Writing it as 24 # was somewhat faster and certainly more legible.

I'm way off track here. Let's get back to equilibrium. Because life is complicated, forces in balance only describes one type of equilibrium — translational equilibrium. Forces left equal forces right. Forces up equal forces down. Forces overall add up to nothing and nothing changes left, right, up down, or at any angle in between. An object is in translational equilibrium if the sum of the forces acting on it add up to zero and it does not accelerate in any direction.

Fleft =  Fright
Fup =  Fdown
F =  0
a =  0

So why do I have to use the modifier "translational"? Because sometimes forces don't push or pull something, they turn it. A push or pull that would result in a rotation is called a torque. Actually, a net torque is that which causes rotational acceleration, but we'll save a proper discussion for another section. When torques are involved, rotational equilibrium is a possibility as well. This too will be dealt with in another section of this book. The graphic above showing "the most perfect balance" is a perfect example of a problem where forces and torques must be dealt with.

The word equilibrium also has a thermodynamic meaning. Thermodynamics is the branch of physics that treats heat (the thermo part) as another form of energy (the dynamics part). Equilibrium in thermodynamics occurs when the internal energy entering a system is balanced by an equal amount of internal energy exiting the system. Net internal energy in transfer from one place to another is called heat. When a system no longer exchanges heat with its surroundings, it's said to be in thermodynamic equilibrium. The image this kind of equilibrium evokes is nothing like placid picture of forces combining politely to produce rest or unchanging velocity. Thermodynamic equilibrium is active and lively. It's an example of a dynamic equilibrium. I used the article "an" because there are others. When the rate of evaporation equals the rate of condensation, that's an example of dynamic equilibrium. When the rate at which nitrogen dissolves into water equals the rate at which it undissolves back into the air, that's another example. Once again, save it for later.

Tests for equilibrium


2 forces

equal and opposite

3 forces


triangle of forces

n forces

polygon of forces