Mechanics

Mechanics Spring 2012 Synthesis Document

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Mechanics by Mind Map: Mechanics

1. Module 1: Vectors and Kinematics

1.1. Coordinate Systems

1.1.1. In discussing the way things move, it's sort of useful to consider the kind of space they move in.

1.1.2. While it would be convenient if math let us consider objects and the space they occupy as they appear in reality, this isn't actually possible. Practically, we'll use some simplified version of reality --a coordinate system. A well-chosen coordinate system greatly simplifies our math, or at least our understanding of the model. A poorly chosen coordinate system will complicate our lives by forcing us to use extra variables or less cooperative operations like the trigonometric functions.

1.1.3. Coordinate systems are partly defined by their unit vectors--vectors with a length of 1 which can be added to reach any point in space we care about.

1.1.4. Cartesian

1.1.4.1. Cartesian coordinates are the ones we're all probably most familiar with. Cartesian coordinates are simple, mapping to the x-y grid we all learned our math on. Most problems with objects moving independently of one another will work well in Cartesian.

1.1.4.2. (x,y, z)

1.1.4.2.1. xi + yj + zk

1.1.5. Polar

1.1.5.1. Polar coordinates make the most sense when we're talking about things that move in circles, ellipses, spirals...helices, along the surfaces of spheres and other round things, and so on and so forth.

1.1.5.2. Because polar coordinates are less familiar, the differences between them and Cartesian coordinates can be daunting. Most notably, velocity and acceleration are almost completely different.

1.1.5.3. (r, θ)

1.1.5.3.1. Rr

1.1.5.4. To convert between polar and Cartesian coordinates we should consider that with a radius and an angle we've defined half of a triangle.

1.1.5.4.1. With some trig, we can convert (r,θ) into and equivalent (x,y) by the relationships x = r*cosθ, y = r*sinθ.

1.1.6. Natural

1.1.6.1. Natural coordinate systems are strange creatures, but helpful for systems with parts moving in relation to one another.

1.1.6.1.1. Rather than basing their units off of an absolute reference frame (ie. (0,0) just floating off in space), natural coordinate systems are based off of the characteristics and limitations of the system itself.

1.1.7. Consider a double pendulum.

1.1.7.1. You could define the position of both weights in a Cartesian system. This allows them to exist anywhere, at any time, relative to one another.

1.1.7.1.1. Any math has to introduce constraints to keep them in place, hopefully involving trig functions over abuse of Pythagoras.

1.1.7.2. You could define the weights in a polar system, with the radius to the first weight being constant.

1.1.7.2.1. The first weight is constrained, it can only move on an arc consistent with the length of its binding.

1.1.7.3. Finally, you could use a natural coordinate system where the first weight uses a polar assignment, and the second is defined similar off of the first.

1.1.7.3.1. Both weights are confined along arcs consistent with their constraints. All motion can be defined in terms of changing angle (and radius direction) and follows intuitively from the problem.

1.2. Kinematics

1.2.1. Kinematics describes the way things move.

1.2.2. As far as we are concerned, this boils down to position (where things are), velocity (how position changes in time), and acceleration (how velocity changes in time).

1.2.3. While these will be controlled by Newton's Laws later on, right now the most complicated thing we'll do will be investigating them in light of the coordinate systems described above.

1.2.4. Position

1.2.4.1. Position is self-explanatory: wherever you go, there you are. Whatever the units define the coordinate system will define your position.

1.2.4.1.1. Position is often represented by d, s, and r, as well as more specific variables specific to a problem (eg. h for height).

1.2.5. Velocity

1.2.5.1. Velocity is change in position versus time. ie. Velocity is dr/dt.

1.2.6. Acceleration

1.2.6.1. Acceleration is dv/dt, or the second derivative of position.

1.2.7. Differentiate things and stuff

1.2.7.1. Cartesian coordinate systems follow Cartesian math: the derivative of a function is just its derivative, taking into account which variables change in time and which don't.

1.2.7.2. Polar coordinates, and natural systems that resemble polar coordinates, do not follow Cartesian math so closely.

1.2.7.2.1. Really, not at all closely.

1.3. Vector Operations

1.3.1. All of these actions are easily defined for Cartesian coordinates. To work in polar or natural coordinates it is usually necessary to first convert in Cartesian coordinates, with the exception of the derivative.

1.3.2. Vector Addtion

1.3.2.1. Vector addition is simple in Cartesian coordinates, add the components together. i to i and j to j makes handy little triangles.

1.3.2.1.1. New node

1.3.3. Dot Product

1.3.3.1. Dot products are the sum of the products of like entries. ie. (2,2,2) dotted with (3,3,3) = 2*3 + 2*3 + 2*3 = 18. Dot products are a scalar product, in that they take two vectors of like length and turn them into a scalar.

1.3.3.1.1. Additionally, and more technically, a dot product is defined as the projection of A onto B.

1.3.3.1.2. This makes the second numerical definition of the cross product A*B = |A||B|cosθ.

1.3.4. Cross Product

1.3.4.1. The cross product is an operation that produces an entirely new vector, perpendicular to the vectors it is performed upon.

1.3.4.1.1. The direction of the new vector is found using the right hand rule, arranging the first and middle fingers along the first and second vectors. The thumb points in the direction of the resultant vector.

1.3.4.1.2. This is possible since any two vectors define a plane (or a line in the case of parallel vectors) and therefore may share a normal counterpart.

1.3.5. Time Derivative

2. Note: This is surprisingly fun.

2.1. Seriously. Actually, really, weirdly, fun.

3. Note: connections are fun.

4. Module 2: F= ma

4.1. Free-body diagram

4.1.1. A useful abstraction tool, for keeping track of the forces on an object.

4.1.2. Simpler compared to other diagrams, since it only tracks real forces.

4.1.2.1. Ignores fictitious forces--things caused by motion.

4.1.2.1.1. eg. Centripetal force

4.1.2.2. Ignores velocity, momentum, energy...

4.1.3. The main trick is to relate forces back to a coordinate system. If you can establish a common set of axes, any force can be broken down into components that either add or subtract from the other forces.

4.1.3.1. Without common axes, forces would interact...painfully.

4.1.4. Often draws on natural coordinates, though many simple problems simply work in Cartesian and (Uniform) Circular motion works well in polar.

4.2. Using the force

4.2.1. Gravity

4.2.1.1. Gravity is one of our most fundamental forces.

4.2.1.1.1. If something is falling or sliding or experiencing friction, gravity is going to show up.

4.2.1.1.2. Force of gravity is equal to mass times acceleration due to gravity.

4.2.2. Contact forces

4.2.2.1. Contact forces spring from Newton's Third law: the law of equal and opposite reactions.

4.2.2.2. If one thing is pushing on another, or just generally being in contact with another object, that's a contact force.

4.2.2.2.1. Contrast with

4.2.3. Friction

4.2.3.1. Friction stems from an interaction between surfaces.

4.2.3.1.1. Individual nooks and grannies on an object's surface can interfere, causing them to stick together.

4.2.3.1.2. Once the initial bind is overcome, the two surfaces glide across one another.

4.2.3.1.3. Friction is proportional to the force between objects, ie. the normal force. This proportionality is represented in the coefficient of friction.

4.2.3.1.4. Stems from the microscopic Van der Waal's force in most circumstances.

4.2.4. Drag

4.2.4.1. Sort of a combination of friction and contact forces in a fluid.

4.2.4.2. Each particle in a fluid interacts with its neighbors, which interact with their neighbors, which...

4.2.4.2.1. Adding an object, which pushes on some particles in a fluid and so on and so forth, we generate chaotic behavior, generally to the net result of slowing down the object.

4.2.5. Other

4.3. Diffie Q's

4.3.1. Mostly we just did basic differentiation and integration!

4.3.2. Integrating F/m, we can find V. integrating V, we can find r.

4.3.3. Conversely, differentiating from values in our system allows us to learn about the accelerations and thus forces of our systems.

4.3.3.1. eg. Pulley systems with ropes on constant length. Defining a relationship between different parts of the system (such as by comparing lengths to either side of a pulley), integration of the constant allows us to establish a relationship between the velocities and accelerations of each part.

4.3.3.1.1. Relating those back to the system as a whole, we can find forces and accelerations and anything else we might care about.

4.3.4. One important thing is the constants of integration are often either the givens or the unknowns in our problems. Knowing a, deriving v = at + v0, we can use the initial condition to find a current v, or a final condition with a known time to find the initial velocity.

4.4. Equations of motion

4.4.1. Newton's Laws

4.4.1.1. Outside forces

4.4.1.1.1. If nothing is acting on an object, its velocity is constant. If it is at rest, it stays at rest. If it is in motion, it stays in motion.

4.4.1.1.2. Defines the use of the second law. No forces, nothing changes.

4.4.1.1.3. Also deals with the application of initial conditions--if something is moving in one direction and takes a force perpendicular to that, the independent motion is unaffected.

4.4.1.2. F = ma

4.4.1.2.1. This looks familar.

4.4.1.2.2. This relationship defines this module.

4.4.1.2.3. If an object has mass and is changing velocity, this law came up.

4.4.1.3. Equal and Opposite reactions

4.4.1.3.1. Constraints

4.4.1.3.2. Normal forces

5. Module 3: Conservation

5.1. Momentum and momentum conservation

5.1.1. Momentum is a frame-dependent, vector quantitiy which essentially defines how in-motion it is.

5.1.1.1. P = mV, where V and thus P and both vectors.

5.1.1.2. We care about relative momentum exactly as we do relative velocity: someone riding a train has no velocity relative to that train, and consequently has no relative momentum.

5.1.1.2.1. While the person in the train sees it to have no momentum, the person in front of the train sees a significant quantity. It is clear why we should be careful how we define our frames of reference.

5.2. Center of Mass

5.2.1. The Center of Mass is a pretty and mathematically consistent lie we tell ourselves to let us work with point objects.

5.2.2. "Center of Mass" essentially represents an averaging of the effect of forces on every particle in a system. Given each particle is rigidly attached to its neighbors, the net effect of the particle's mass on the behavior of the system as a whole is given by r*m.

5.2.2.1. Consider levers: the longer the arm, the greater the change exerted by the same force.

5.2.3. Summing every possible point of mass, we see an integral taking shape, taken across an object (easy in one dimensional objects, more complicated in two or three dimensions) in dm.

5.2.3.1. All possibilities of fourth-dimensional Centers of Mass have been suppressed by the Department of Incredibly Bad Ideas.

5.2.4. Any force acting on an object does some work to its center of mass. If all we care about is the general motion of an object, we can reasonably model it as a point mass by just tracking the motion of its 1-D CoM.

5.2.4.1. While gravity is well-behaved and acts pretty much exclusively on CoM, other forces don't. To model the rest of an object's motion, we have to take into account Torques and other concepts.

5.2.4.1.1. Since only very silly people would ever want to do this, I can reasonably say there is no reason--ever--that this document should take this into account.

5.3. Impulse

5.4. Conservation of energy and the work-energy theorem

5.4.1. "Conservation of Energy" is as fundamental as it is easy to define. In the normal state of things, with no forces acting on an object, energy is constant.

5.4.1.1. Certain forces are nonconservative and are said to do work.

5.4.1.1.1. W = E1 - E2

5.4.1.2. Friction is a common nonconservative force.

5.4.1.3. In defining conservative versus nonconservative forces, we have to consider our system.

5.4.1.3.1. A block moves across a fricitonless surface, until it collides with a spring. Then, the spring and friction begin to slow it down.

5.4.1.3.2. Relative to the system as a whole, we see see both conservative and nonconservative forces at work.

5.4.1.3.3. Relative to the block, both forces are nonconservative, taking energy out of the block.

5.4.1.3.4. To calculate the work off one force, given the other, we can find the total change in energy of the block and find the difference.

5.5. Potential energy and conservative vs. non-conservative forces

5.5.1. Conservation of momentum does not imply conservation of energy!

5.5.1.1. Both elastic and inelastic collisions conserve momentum without conserving energy.

5.6. Collisions and other multi-object systems

5.6.1. The most common interaction between several objects we care about is the collision.

5.6.1.1. We know from experience that two objects colliding transfer some energy. In the simple case where they collide with no other interactions there are two types of collision.

5.6.1.1.1. A third type--super-elastic--occurs when more energy comes out of the collision than went in. Consider springs; also: flubber.

5.6.1.2. Elastic

5.6.1.2.1. Elastic collisions preserve momentum of the system.

5.6.1.2.2. Elastic collisions preserve kinetic energy of the system.

5.6.1.3. Inelastic

5.6.1.3.1. Inelastic collisions preserve momentum of the system.

5.6.1.3.2. Inelastic collisions DO NOT preserve kinetic energy

5.6.2. Another case is during mass transfer.

5.6.2.1. If we consider the case of men leaping off of a train car--assuming the car isn't so massive as to make their efforts negligible--obviously some momentum change is taking place.

5.6.2.1.1. It is possible the men are jumping in front o fht ecar, adding a positive dm/dt term with a delay of a few time steps. We neglect this case.

5.6.2.2. All mass in the system has momentum. After some amount of mass leaves the system mv must have changed. The question is how and how much.

5.6.2.2.1. We can expand F = ma to F = dp/dt

5.6.2.2.2. In this case, as both velocity and mass change in time, we see F = dm/dt*v + m*dv/dt.