Monday, August 6, 2012

Acceleration, Energy, and Momentum

In any Dynamics class, 3 main methods of analysis will be taught -- Acceleration, Energy Methods, and Impulse/Momentum. Within the class, it’s pretty easy to figure out which method the professor wants: If we’re in Chapter 4, it must be Energy, right?

What about later on? While I’m still only through the first-level Dynamics course in my curriculum, I do assist others with the class occasionally (I’m a walk-in tutor for some of the 200/300 level engineering courses here), and haven’t yet developed a good set of rules to delineate between the methods. Here’s a first effort, hopefully I can add more in the future.

Acceleration Methods: Newton’s Second Law

Newton’s Second Law, F = ma, is the foundation for the entire discipline of Statics, and as such is a good transition from a Statics course into Dynamics. As such, the problem-solving steps here are very similar to what a statics student will be accustomed to:

  • Draw a Free Body Diagram
  • Sum the forces in each orthogonal component
  • ?
  • Profit!

Of course, we now have moving bodies to worry about, so the missing step is to draw a Kinetic Diagram of the object in question. This simply involves drawing the applicable velocity and acceleration vectors. Then, we can equate our acceleration components to our force components and solve the problem. 

This gets a bit tricky in different coordinate systems, but it’s the process that counts.

So when are Force/Acceleration methods the right tool for the job? Well, since we’re relating force to acceleration, take a look at the information given: If we have several known forces acting throughout a time interval on an object, or have an easily described acceleration, then F = ma will be easy to solve. Or at least possible to solve.

Energy Methods: Newton’s Second Law, Over a Distance

Perhaps ‘easy’ isn’t the correct word, or perhaps we’re more concerned with efficiency of the solution. Take, for example, a bungee jumper. 

In this case, the force applied is a function of the distance travelled, which yields a 2nd order-ode. Since ads = vdv, we can still solve by direct integration. But what’s the broader principe here? What have we actually done? Essentially, we’ve integrated both sides of F = ma
  • On the left, Force over a distance, yielding Work.
  • On the right, we do some chain rule work to end up integrating the velocity term. Acceleration times a distance is equivalent to velocity times a change in velocity, right? Anyway, this results in 1/2mv^2, a term we call kinetic energy.

Note that this is not at all rigorous, and skips some pretty important stuff which I will hopefully cover later. For now, though, the gist of the idea: Work causes a change in kinetic energy! A simple idea, but one that we retrieved simply by manipulating Newton’s second law. The principles also extend to potential energy, such as the gravitational energy our jumper above benefits from.

More importantly, this is a template we can use to talk about changes in energy as a result of work done on an object, or vice versa. Rather than drawing the traditional FBD and labeling forces (it’s still good practice to do this anyway), we can simply draw an Energy Diagram of the situation at large. So, in the case of the jumper, there are a few points of interest:

  1. Standing on the platform. Lots of potential energy, no kinetic energy.
  2. Cord begins to stretch. Still plenty of potential energy, and a good amount of kinetic energy.
  3. Wherever the jumper stops. (Hopefully above the ground--we haven’t reached impact analysis yet!) Kinetic energy is 0, potential energy is either neglected or small, and ALL of that energy has been transferred to the cord. At this point, our analysis stops: Simple work-energy principles aren’t sufficient to, say, determine the ‘snap-back’ of the cord. 

So that’s where the limitations of Work-Energy show up: We can only find out broad characteristics of an object at different points along its path. Since the conservative forces we model are ‘path-independent,’ we actually can’t find out anything about the path of the jumper--only make an estimate of how far he falls before stopping.

But sometimes, a quick-and-dirty feasibility analysis is what’s needed. Work divided by time is power, so Work/Energy gives an easy way to put upper and lower bounds on power requirements for a motor, for example. Roller coaster problems are also classic examples of the method: ‘How tall a drop to get this coaster around this loop?’ 

Momentum/Impulse: The 2nd Law Over Time

Finally, we have a class of problem that occurs very often in real life, but can’t be adequately modeled with either of the above methods: Short, violent impacts. Car crashes, baseball bats, a sprinter’s foot...the examples are endless. The problem faced in these situations is that very complicated, volatile accelerations can occur over very short time intervals. Attempting to solve and integrate large, discontinuous acceleration will give unreliable results.

So, we’ll again integrate Newton’s 2nd Law, this time with respect to time. 

  • On the left side, a force acting for some amount of time is called an impulse. 
  • On the right side, integrating acceleration with time is velocity. We call mass times velocity momentum. When people say ‘inertia,’ they’re generally speaking of ‘momentum.’ (i.e. “The truck has a lot of inertia at 60 mph.’) 

So what this gives is a way to relate impulses directly to changes in velocity. Additionally, we can analyze collisions between objects, if we know their velocities before or after the collision. 

Anytime you’re presented with quick, violent forces and want to look at how they change the velocity of something, use impulse/momentum. 

Momentum also leads into more advanced mechanics concepts, such as angular momentum, and allows analysis of mass flows (fluids) and variable mass systems (rockets). 

My Dynamics textbook is Engineering Mechanics: Dynamics by Gray, Costanzo, Plesha. It hasn’t received good reviews on Amazon, although the 2nd edition is better-organized than the 1st edition I own. I think the ‘Gold Standard’ of Dynamics texts is anything by Hibbeller, and old editions can be had quite cheaply, if you’re so inclined.

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