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

The purpose of this section of this book is to serve as a repository for problems in dynamics. The acceleration in every problem will be nonzero in one direction. 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.

The neat force

Take Newton's first law of motion and break it into two parts. "An object at rest tends to remain at rest and an object in motion tends to continue moving with constant velocity…." That long main clause is where statics lives. "Unless acted upon by a net external force." This short subordinate clause is where we find dynamics.

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). This can be written as F (using the Greek letter sigma to indicate a sum and boldface to indicate that forces are vectors) or as Fnet (using the subscripted word net to make the symbol read more like spoken language and italics to indicate that knowing the magnitude of the force is often all that matters) or other variations like these.

Force is a vector quantity, which means that direction matters. Use positive values for forces that point in the preferred direction and negative values for forces that point in the opposite direction. If a problem is two dimensional, pick two preferred directions at right angles — something like up and to the right. Pick preferred directions that make your life easy. The laws of physics do not care if you call right positive or left positive. Space, in the mathematical sense, is isotropic. It measures the same in all directions.

Newton's second law of motion describes how net force, mass, and acceleration are related. Basically, net force causes acceleration and mass resists it. The best way to write that is not with words but with symbols. Something like this…

F = ma  or  Fnet = ma

You are now ready to begin the next phase of your training.

For example

Take the unexceptional example of an unexceptional bicycle being unexceptionally pedaled down an unexceptional flat, level road in an unexceptional manner. What are the forces acting on the bicycle and rider (together as a whole)?

Start with the obvious. Everything has weight and weight points down. The bicycle is on a solid surface so there's a normal force pointing normal to that surface. The surface is level so the normal direction is up. The rider is pedaling. This means there's some kind of force pushing the bicycle forward. I don't want to overanalyze the situation, so let's just call that force push. Even properly inflated tires resist rolling, the axle may or may not need lubrication, and the air certainly drags on a moving body. Let's make life simple and call all these forces together friction. The rider pushes the bicycle forward and friction pushes backward.

We're ready to make a free body diagram. Draw a box to represent the bicycle and rider. Draw four arrows coming out of the center of the box to represent the four forces acting on the bicycle and rider. Although not always necessary, one should try to draw the arrows with lengths that correspond to the relative magnitudes of the forces. Long arrows for strong forces. Short ones for weak ones.

Start with the easy pair — weight and normal. Nothing is happening in the vertical direction in this scenario. The road is level and the rider isn't performing a stunt. Weight and normal will cancel each other out. Draw one arrow down and another one up and give them the same length.

Finish with the slightly less easy pair — push and friction. Something is happening in the horizontal direction. Motion is happening in the horizontal direction. The bicycle is going somewhere. That should be good for something. Shouldn't it?

Sorry, but no. Motion isn't what matters. Change in motion is what matters. Is the bicycle accelerating or moving with a constant velocity? Acceleration makes a situation dynamic. Lack of acceleration makes it static. The direction of the net force determines the acceleration. The force pointing in the direction of the net force will be the stronger of the two.

If the bicycle is speeding up, then the rider is pushing the bike forward more than friction is pushing it backward. If the bicycle is moving with a constant velocity, then push and friction are equal. If the bicycle is slowing down, then the force of friction wins over the force pushing the bike forward.

Here's a bunch of drawings that show what I just said.

Three free body diagrams

Here's a bunch of equations that show what I just said. I like to use up and right as the positive directions, but that is not a law of physics. It's just a preference.

ΣFy  = 0
N − W  = 0
N  = W
ΣFx  = ma
P − f  = ma
speeding up
P  > f
a  > 0
constant speed
P  = f
a  = 0
slowing down
P  < f
a  < 0

Go now and solve problems.

Wrapped (warped) coordinates

two bodies connected by a string