Built on Facts

This is Zhang Juanjuan, immediately after applying an impulse to an arrow. Impulse is something which gets less airtime than work in freshman physics, but it’s nonetheless very important. It’s sort of momentum’s version of work.

You’ll remember from Monday that there are two main things to keep in mind about work.

1. Work is defined as force through a distance
2. Work equivalent to the change in energy

You can think of impulse the same way.

1. Impulse is defined as a force thorough a time
2. Impulse is equivalent to the change in momentum

Crucially, impulse is a vector and work is not. Force is a vector because it has a magnitude and it’s being applied in a particular direction. But work isn’t a vector because it doesn’t have a direction – it’s just an amount of energy. Mathematically speaking, the dot product of two vectors is not itself a vector, it’s just a number.

Impulse however is a force vector multiplied by elapsed time, which is just a number. A vector times a number is another vector. So when Zhang Juanjuan fires that arrow for the gold medal, she has done work (the force of the string through its distance of travel) and applied an impulse (the force of the string during the time it took to travel that distance).

Now let me get some mathematical justification out of the way. The force might not be constant, so the total impulse is the sum of each bit of force times each bit of time during which it acted. That means by definition impulse is

But via the definition of force as the change in momentum with respect to time, we have

Still with me? If so, you might be curious as to why anyone would bother with this concept. The main reason is that it allows you to not worry about the details of the force as long as you can say something about the change in velocity. For an easy example, a baseball batter hitting a ball has applied a tremendous force over a very small time and distance, so the details of the work are difficult to calculate. But the initial and final velocities of the ball are easy to see, and so you can work backwards to find the impulse and from there find something about the forces involved. You’ll also see an extension of this concept called specific impulse used in rocketry to quantify how much momentum change you can get out of a given quantity of fuel.

Just as work presages a rewriting of Newton’s force mechanics into an energy-based form, impulse can be thought of as a first step in writing Newton’s laws in their Hamiltonian form. The Hamiltonian itself is the total energy expressed in terms of a generalized version of momentum.

Comments

#1 Jennifer Ouellette
August 20, 2008

Dear Matt: your blog totally rocks. I swear, I end up clipping and saving half the posts for future reference…
#2 Uncle Al
August 20, 2008

A Remington .357 magnum 125 gr. (8.1 grams) semi-jacketed hollow point bullet goes from rest to muzzle 442 m/sec (Mach 1.3) in 11.4 cm (4.5″). 790 joules imparted.

Average velocity would then be 221 m/sec through 11.4 cm or a 0.516 msec traverse. v_(final) = at; [(442 m/sec)/(0.516 msec)]/9.8 m/sec^2 or 87,400 gees average. Impulsive!
#3 Zifnab
August 20, 2008

You need to change the title to “Built on Math”.
Moar plz.
#4 AtheistAcolyte
August 20, 2008

This may be a remarkably idiotic question to ask, but I figured I may as well bring it up: Why is it that “work” happens when I hold an object at arm’s length, but no work really happens?

I understand there is probably confusion between my colloquial “work” and physics-style work, but energy is certainly being expended, and force is being applied. I’ve tried looking at the muscle tension as the source of work, but I still don’t quite know how to describe that mathematically. Being a physics/math dude yourself, I thought you might have a good answer for me.

Again, sorry if this makes me sound ignorant, but searches on the intertubes have been somewhat schizophrenic in their responses.
#5 Matt Springer
August 20, 2008

The biochemistry of muscle tissue is not something I know too much about, but my rough understanding is that each muscle fiber actually twitches to contraction and then releases and the process starts over. Each twitch does work and requires energy, but there are so many fibers that the many individual twitches average to what seems like a contracted muscle standing still. It’s like a choir holding a very long note – you can’t tell when an individual stops to take a breath because at any given time most of them are still singing.

There’s a pretty good discussion in a bit more detail in the comments of the post about work a couple days ago.

Also, thank you everyone who has had nice things to say about the site! There’s not much that’s more satisfying than hearing that other people enjoy this stuff as much as I do.
#6 fok
August 20, 2008

So if Hamiltonian mechanics isn’t the energy based form what is? Are you claiming that title for lagrangian mechanics, since it’s based on q and \dot{q} instead?

Perhaps actual physicists would agree, but I don’t think I do.
#7 Matt Springer
August 20, 2008

Hamiltonian mechanics is definitely formulation of dynamics in terms of energy. “The Hamiltonian” is the energy, after all. The role that momentum plays just happens to be more direct in Hamiltonian mechanics than Lagrangian mechanics.

The main point is that all these formulations are just mathematical restatements of each other: Newton in terms of force, Lagrange in terms of classical action integrals, Hamilton is terms of total energy itself expressed in terms of generalized momenta.
#8 natural cynic
August 21, 2008

…each muscle fiber actually twitches to contraction and then releases and the process starts over. Each twitch does work and requires energy, but there are so many fibers that the many individual twitches average to what seems like a contracted muscle standing still.

In the case of holding something in a static position, the muscle force is produced by recruiting number of motor units – single nerve fibers that innervate a number of muscle cells. There is constant feedback through the sensory system [detecting force and position] to maintain the proper number of motor units recruited. The motor units release acetylcholine fast enough so that each muscle fiber maintains force until fatigue – at which time the nervous system will detect decreasing force and recruit more motor units to maintain the weight’s position. Twitches never really happen, stimulation cannot be turned on and off that rapidly. What happens is that twitches blend together into a tetanization, which will be maintained until fatigue of nervous stimulation is stopped.

It’s like a choir holding a very long note – you can’t tell when an individual stops to take a breath because at any given time most of them are still singing.

This is what happens on the level of myofibrils – the proteins where myosin filaments interact with an actin filaments to form crossbridges and one filament slides over another using ATP to produce a contraction. The crossbridges are still being broken and reformed as isometric force is being applied even though no movement takes place. Since ATP is continually being used, chemical “work” is being done.
#9 CCPhysicist
August 21, 2008

Thanks, cynic. Its that term “crossbridges” that I had forgotten from my HS biology class about muscles.

As to the side remarks about alternative formulations of mechanics, the important detail that did not get stated is that you need BOTH the energy and momentum equations (in either their integral or differential form) to reproduce what you get from Newton’s equations. Newton relates space and time to each other directly through forces. Energy and momentum give you a way of finding out about one of them even if you are ignorant of, or don’t care about, the other.

PS –
The jump back and forth between the integral and differential form of various laws (in mechanics or electricity and magnetism) is “just” the mathematics of the Fundamental Theorem of calculus. Sadly, math faculty seem totally unaware of the conceptual significance of the fundamental theorem outside mathematics and fail to give students a reason to learn (retain) it.
#10 Sophie Doucet
August 24, 2008

so, I have really enjoyed reading your posts Matt. You never mentioned a girlfriend, so I was just wondering if you were taken. If not, maybe we can talk about science sometime.
#11 Matt Springer
August 24, 2008

Well, I never thought I’d get flirted with flirted via a physics blog. I’m very flattered! But I do have a very wonderful girlfriend who I would not trade for the world, so to all you interested ladies out there (ha!) I must decline.

Besides, there’s a lot of good single guys in my department. They’d be happy for a date!