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Because point masses are hard to come by, and because real objects are not points, you have to do a lot of vector addition. Fortunately there is one important special case that has a simple result: the sphere. A spherical object exerts a force on another mass (OUTSIDE THE SPHERE) that is the same as if the sphere were a point. This requires a bunch of integration to prove. Since the Earth is almost a sphere (and the sun too), we can conveniently do most of the analysis in this approximation.
For any system with several masses, remember that force is a vector! For dealing with potential energy, recall the basic defining equation for potential energy (and no it is not mgh)

F_x = – dU / dx or F_r = – dU / dr

When we look at the motion of planets around the sun, it's still F = ma, and in this case the orbits are very close to circles, so the acceleration will be v²/r toward the center. In other words you will be reviewing and using all the old material, just applying it in a new context.

ch 13: Equation (13-8) is central to this topic. It appears in various guises, as the equation (13-33), and approximately equation (13.42), but it's always basically the same. That means its solution is always the same even if the symbols are very different. Rather than use a systematic approach to solving this equation, just use your experience with calculus to guess the form of the solution and then adjust the parameters until it works. There are various forms for the solution:

x = A cos( ωt ) or x = A sin( ωt ) or x = A cos( ωt + φ ) or x = A cos( ωt ) + B sin( ωt )

All of these work, and the third and fourth ones include the first two as special cases. ( φ = 0 and φ = –π/2 respectively )
The subject of forced oscillations and resonance appears in section 13-8, and they give a pretty good qualitative presentation. Because they want to include the damping forces the mathematics would get a lot more complicated so they just present the results. If you take the equation (13-52) and set the damping factor "b" to zero, you get something that looks more like what I did in class.

ch 9: Some of this chapter is simply a repeat of chapter two. The difference being that instead of the single coordinate being "x" for the displacement along a straight line, it is the single coordinate "θ" for the angle of rotation about a fixed axis. Previously for the unit of distance you could use a wide variety (meters, light-years, feet) and if you pick an unorthodox unit you simply had to handle the conversions. For angles, the radian unit is singled out as distinct because all of the formulas that use calculus assume that unit and no other.
Equations (9-2) and (9-7) (without the vectors) are the analogs of (2-10) and (2-14), and everything that follows for x follows for θ. You can do the same sort of antiderivatives (or integrals) with respect to time as before.
An important relation between angular motion and linear is v = rω. That's what you need to get the equation for the kinetic energy of a rotating rigid object.

ch 7: Conservation of momentum is always valid for a system where the total external force is zero. It remains a very good approximation in a collision if the external forces are small compared to those from the collision (and if as usual, the collision itself takes a short time).
Kinetic energy is usually NOT conserved. Only in the special case of a completely elastic collision can you use this sort of conservation equation.

ch 8: Conservation of momentum is always valid for a system where the total external force is zero. It remains a very good approximation in a collision if the external forces are small compared to those from the collision (and if as usual, the collision itself takes a short time).
Kinetic energy is usually NOT conserved. Only in the special case of a completely elastic collision can you use this sort of conservation equation.

ch 6: The general definition of work is not just an integral of F_xdx. What if you're moving in three dimensions, not just along a straight line? The answer is involved, and the author pretends to answer it, but I think doesn't help. See equation (6.16) for what I mean.

Instead of looking at the most general case, there are really two special cases that are worth learning. The first uses the integral definition for work (F_xdx) in one dimension. The second special case is for three dimensions, but for a constant force. In that case, work is the dot product F^.Δr. Sort of "force times distance," but specifically a scalar product. In both cases, the work-energy theorem says that the total work done on a mass gives you its change in kinetic energy.

ch 7: The core equations are buried in the middle of the chapter, equations (7-5), (7-7), and (7-18). The only non-conservative force we encounter is friction in one form or another.

ch 6: For the start, concentrate on one dimensional stuff, where the basic equation is the combination of (6-6) and (6-14). When the force is a function of position instead of a function of time, the methods that we've used up to now fall flat on their face.

W = ΔK is the most abbreviated summary of the equation.

Work is not "force times distance" except in the most special case that the force is a constant. That's exactly like saying that the equations resulting from constant acceleration are special cases (at²/2 instead of a_x = d²x/dt²). The general expression for work is an integral from the initial position to the final position of F_x with respect to x. Equation (6-14) for example. Note: "with respect to x," not "with respect to t."
Power is the rate of doing work, and so it is a derivative: dW/dt as in equation (6-18). Because a little bit of work causes a little change in the kinetic energy, this power is the change in the kinetic energy too.
Watch out here! When you say F = ma, you really mean the total force. The same for work. The W in the equation W = ΔK is the work from the total force.

ch 8: This chapter is both easier and less complicated than chapters 6 and 7. We'll return to those shortly.
Section 8-2: Ignore impulse and (for the moment) energy.
The definition of momentum is p = mv, and the basic equation in Newton's second law is F_total = dp/dt. If the mass is a constant, this is the same old F_total = ma. If m isn't constant then the new form is the right one. This won't happen too often for us, but it's nice to see it stated right.
Conservation of Momentum is one of the major results in mechanics. "Conservation of" has a special meaning. It means that if you evaluate the quantity now and evaluate the same quantity later, you can place an equal sign between them. The authors commonly use a capital letter P to distinguish total momentum of many particles from the lower case p of one. This is not standard notation and I recommend that you don't use it --- how do you distinguish a P from a p when you write them? They do the same thing for acceleration of the center of mass. Again, I've not seen this done anywhere else and I prefer to use a subscript such as _cm to designate center of mass.
Skip section 8-4, 8-5, 8-7, 8-8 for now.

ch 5 again: What is tension? That's the magnitude of the force that a rope will pull with. If the rope is very light, it will be uniform through its whole length. Ropes can't push.
Example 5-6 shows a couple of things. When you have two masses you just do everything twice. You figure out the things that act on each and apply F = ma to each. It also shows another idea: When you're treating the two masses as two separate problems to set up the equations, there's no requirement that you use the same coordinate system for both. In this example one mass is more easily described by a tilted coordinate system and the other by a vertical one. It makes the problem a lot easier.
Friction can be complicated, but there are a couple of pretty good approximate expressions that describe the subject adequately. Those are all that we will use. When two objects are in contact the force that one exerts on the other is not necessarily perpendicular to the surface of contact. It usually isn't, and that's called friction. The author uses the notation "F_N" for the perpendicular component of the force ("normal"). The magnitude of the tangential component of the force then obeys one of the two relations

F_fr = μ_k F_N (kinetic or sliding case)

F_fr < μ_s F_N (static case, no sliding)

The normal component of the contact force isn't necessarily from gravity, so don't assume that it is.
All the problems in this chapter are set up the same way. If you take shortcuts, you can readily turn an easy problem into a difficult one.

ch 5: For setting up problems in mechanics, there are steps that make it straight-forward. If you attempt to invent a new procedure for every problem, you'll drive yourself crazy. Instead follow a standard procedure and the problems are much easier.
1. Sketch
2. List things applying forces to each mass involved
3. Choose a basis (i, j) and write the total force on each mass in this basis
4. Determine what you can about the acceleration of each mass and write it in the same basis
5. Apply F_total = m a to each mass
6. Break into components
7. See if any more equations are needed and if so, find them. These will commonly be relations between how different masses move (easier to see in examples). Don't forget Newton's third law!
8. Solve simultaneous equations
9. Check results

ch 4: a = F_total / m
It sounds easy, but this chapter may even challenge your intuition.
Do you have to apply a force in order to keep something moving? (No.)
If a heavy person and a light person try to push each other around does the heavy one exert more force than the light one? (No.)
The first of these statements is contradicted by Newton's first law of motion. The second statement is contradicted by Newton's third law of motion.
In trying to figure out what a force even is, it will help immensely if you remember that in this course, there will be two and only two types of force that you will encounter:
(1) gravity (commonly from the Earth)
(2) contact (another object is touching the first)
There are other types of force, such as nuclear and magnetic, but not here.
The explanation of "inertial frames" in section 4-1 sounds abstruse, but it really only defines a context. If no force acts on an object then its acceleration is zero. Unless this is true then you certainly can't have a = F_total / m. If you are on a carnival ride, going in circles and being bounced around, you can say "I am the center of the universe and the world is going around me." It's legal, but it's not an inertial frame.
When you start to apply Newton's laws you will first have to figure out what things are acting on each object you deal with. Is the Earth pulling on it (gravity)? Is something in contact with it? That provides another force. These are all vectors, so you will usually write them in components and manipulate the equations in that form.

ch 3 again: The examples of projectiles in section 3-4 make it look as if there's a lot to learn here. There isn't. Because gravity provides just a constant acceleration along one direction you can simply break apart the basic defining equations for velocity and acceleration into their components.

v_x = dx / dt, a_x = dv_x / dt, v_y = dy / dt, a_y = dv_y / dt

For gravity you typically take the y-axis to be up (or down) and the only acceleration there is from the gravitational field, g. From that point each coordinate behaves just as in the one-dimensional case.
Circular motion is different, as the acceleration isn't constant. It may have constant magnitude but certainly not constant direction. You can almost get the answer for this case by checking the units: Is there any combination of speed and radius, v^ar^b, for some a and b that even has the units of acceleration? There is only one, v² / R. We will use this result a lot.

ch 3: This combines chapters one and two. Again the basic concepts are velocity and acceleration, but now they're vectors.

v = dr / dt a = dv / dt

I'm not putting arrows over the vectors, but simply using boldface to indicate a vector. That's the same notation as in Spiegel's Vector Analysis book that you are using, and my reason for this is simply that it's a lot easier to write boldface text than it is to put an arrow in place when you write in html. The definition of a derivative remains the same as it was before, but the numerator is a vector. Equations (3-5) and (3-15) are the central equations in this chapter.
The position is not a single coordinate, but a vector, denoted r or r( t ). It has components that are just the x and y coordinates (or x, y, and z if you need all three). That is

When you differentiate this, the unit vectors are constants, so all that changes with time are the coordinates x, y, z.
The techniques for solving problems, at least at the start of the chapter in sections 3-1 through 3-4, are the same as in chapter two. You will commonly know the acceleration (maybe it's from gravity), and you can take the anti-derivatives of each of the components of the acceleration just the same way that you did when there was only one component to worry about.
There are a bunch of equations about projectile motion. None of them are fundamental because they all follow from the anti-derivatives of the (constant) acceleration.

Draw Pictures. Draw Pictures With Arrows.

ch 2 What is the definition of velocity? Of acceleration?

v_x( t ) = [ x( t + Δt ) - x( t ) ] / Δt in the limit that Δt goes to zero

From this you can compute the velocity for simple cases such as the quadratic polynomial that I did in class. Then learn the common answers for a few other cases. This is the derivative.
The anti-derivative is easy once you know the derivative, and in this course it's what you see most often. The anti-derivative is the function whose derivative is what you've got. In class I showed that

That means that the anti-derivative of 2Ct is Ct² + D. There's no guarantee that the constant is the same. This is how you go from a situation where you know something about the velocity to knowing something about the position. Also, going from knowing the acceleration to knowing the velocity is done the same way.
Most problems in chapter two is done this way. You start with the acceleration and take a couple of anti-derivatives to get the velocity and the position functions. Then evaluate the arbitrary constants that these produce by using more information from the original problem.
Don't be afraid to introduce another variable in order to describe a problem mathematically. That's often the easiest way to set up the algebra.

There's a notation that's used in our text that I don't like. When they write something such as 15 kg/m·s do they mean (kg/m)·s or kg/(m·s)? They should use parentheses but they don't. When you see this, assume that they mean kg/(m·s), because if they meant the other, they could have written kg·s/m.

ch 1: Pay attention to the units and dimensions. This is not just being fussy, it becomes a technique that is one of the more valuable tools in your arsenal.
For significant figures, be reasonable. Just because a pocket calculator tells you that the speed of a car is 45.0387492 mph doesn't mean you should believe it. Do you really know it to one part in ten million?
Converting units is always the same thing. You multiply by one. The first set of practice problems for the quizzes is on precisely this topic. You may as well get it behind you early. Those notes expand on the text section 1.2
Section 1.6 starts the central subject of vectors as a geometrical idea. This will be critical in everything we do for the entire course. They then introduce some computational tools in the same section: Components are numbers (typically with units), and they provide a way to translate what can be complicated geometry into much simpler algebra. It's the same idea as coordinates in analytic geometry, but applied to a somewhat different object.
The first few pages of Spiegel's text on Vector Analysis has a concise summary of this information, and the examples flesh it out. You should understand examples 1-9 in chapter one of Spiegel.

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