I know that momentum = mass * velocity, but what does that mean exactly. Usually, when I pose this question, people give me the definition for inertia. Is it just a name for mass * velocity? For your next trick, you can explain angular momentum and then explain spin.
Thanks for your help,
Rob
You may be looking for something more technical than my answer…but perhaps it would help to think of it this way:
momentum = the force required to stop an object’s motion along its current vector.
So, the momentum of a baseball pitch in flight is equal to the amount of force you’d have to apply in the opposite direction to stop it in its tracks.
Help any?
And while we’re at it, can someone explain this idea I’ve always found confusing:
a photon does not have mass, but it does have momentum.
Unfortunately, I think that that is the definition of inertia. Momentum is not force.
Photons have momentum because they have mass, but not rest mass.
Thanks,
Rob
Inertia is the abstract tendency of a body to carry on doing what it’s doing (as per Newton)
Momentum is the measure of force required to change what a body is doing.
Inertia is a property - a scientific principle. Momentum is a quantifiable value.
Momentum can certainly be considered a force. A potential force, but a force nonetheless. It’s kinetic energy.
Inertia is a measure of a body’s resistance to acceleration, which is not the same as what I was talking about.
[on edit, I see that **Mangetout ** has put it more succinctly…and probably more correctly…]
Figaro is correct in his explanation. Force equals change in momentum over the change in time. It is, therefore, the derivative of momentum with respect to time. This makes momentum the integral of force over a period of time, which, of course, is what Figaro was saying.
Momentum is represented by the simple equation p = m * v. It is, of course, a vector. It is also frame of reference dependent, as a result.
Inertia, by comparison, is a less easily defined concept. Perhaps your confusion stems from not being sure what meaning is attached to inertia?
How can momentum be a force or kinetic energy? We already have equations for them, F = ma and E = 1/2 mv[sup]2[/sup]. Momentum is in different units. Force is Kgm/s[sup]2[/sup] and energy is Kgm[sup]2[/sup]/s[sup]2[/sup].
Momentum is a property of an object. It is indeed the product of mass and velocity. It is a conserved quantity, meaning that , in order to change momentum, you must apply force. In many ways, momentum seems more fundamental than velocity – the ohoton has momentum, but no rest mass, as you’ve observed, so perhaps you can think of velocity as momentum divided by mass.
The conservation of memontum isn’t just cosmic bookkeeping – it’s inextricably tied into the symmetry of the situation. If position doesn’t explicitly show up in the system Lagrangian, then momentum will be conserved (If something can happen at x1 or at x2 and there’s no difference, momentum is conserved). One of the uncertainty relationships is that position and momentum cannot be simultaneously known to arbitrary accuracy (Heisenberg’s uncertainty theorem). This has nothing to do with Observer interaction and everything to do with the wave nature of things.
And, yes, when you give the definitions of inertia, they sound almost exactly the same as descriptions of momentum – tendency for things to remain in the same state of motion unless acted on by an outside influence.
Angular Momentum is extremely similar – it’s the tendency of things to continue rotating at the same angular velocity about their center of mass. In the absence of extrenal influences, they will continue to do so. To change angular momentum requires the application of Torque (which is Force applied X moment arm length). The same sort of symmetry relationship holds for angular momentum – if angle doesn’t show up explicitly in the Lagrangian, angular momentum is conserved (or, as above, if the same things can happen at two different angles, the conservation law holds). There’s a Heisenberg relatiuonship for angular momentum, too.
Angular momentum is the product of an object’s Moment of Inertia (represented by I) and its angular rate of rotation. You get I by adding up all the contributions of mass elements times the square of their distance from the axis of rotation. Spin is intrinsic angular momentum of elementary particles.
Momentum is mass times velocity. (However, see below). But you seem to be looking for more than this, so let me suggest that what you really want to know is “Why is momentum important enough to have a name?” I mean, I could take some quantity like mass cubed divided by velocity, and call it ‘Timmervation’ or something, but the name would never catch on, not only because it’s silly but also because nobody really needs a special word for mass cubed divided by velocity – that’s not a quantity that comes up very often, so when it does you can just say “mass cubed divided by velocity”. Whereas, mass times velocity comes up all the time, so it makes sense to give it its own name.
Why does momentum come up all the time? The short answer is that it’s a “conserved quantity” – i.e., the total momentum in a system won’t change, provided that certain conditions (which I’ll get to in a second) are met. Not surprisingly, knowing that certain quantities won’t change over time can be useful in predicting how a system will change over time.
To understand when momentum is conserved, just look at Newton’s second law. It’s usually written F=ma, but a somewhat more general form is “The sum of the forces equals the rate of change in momentum.” Because internal forces cancel out, we can just take this to be the sum of the external forces. (E.g., try jumping into the air and then shoving yourself to one side. It won’t work, because as your hands push your body one way, your body pushes your hands the other way, leading to no net push.) So if there’s no net external force on the system, then Newton’s law tells us that the rate of change in momentum is zero – in other words, momentum is conserved.
Regarding angular momentum: Just like Newton’s law tells us that the rate of change in momentum equals the total force (i.e., the total “push” or “pull” on the system), we can define a quantity whose rate of change equals the total torque (i.e., the total “twist” on the system.) This quantity is angular momentum. It’s conserved when the net external torque is zero. You can also define angular momentum as angular velocity times momentum of inertia. “Angular velocity” is the rate of rotation, and “moment of inertia” (which isn’t the same as “inertia”, by the way) is a measure of the object’s resistance to torques, in much the same way that mass measures the object’s resistance to forces. So this is just the rotational equivalent of “momemtum equals mass times velocity.”
Spin (I assume you mean the quantum mechanical property called spin) is somewhat trickier. As I said, under certain conditions angular momentum is conserved. However, when dealing with atoms and electrons and such, we have to add in an additional angular momentum that is intrinsic to the particles themselves. Let me try to clarify with an example. Different atoms absorb different frequencies of light. The frequency of light is related to its energy, so the fact that atoms only absorb certain energies shows that they have certain discrete energy levels, and thus can only absorb an amount that equals the difference between two levels. But you can’t go from one level to any other level – rather, there are certain transitions which are “allowed” and certain ones which are “forbidden”. These rules are called “selection rules”, but they’re really just conservation laws. My point is that we can look at what frequencies of light an atom absorbs, and deduce something about what it’s conserved quantities are.
I’m trying to link spin into the above discussion in a straightforward way. However, at the risk of getting off track, I feel obligated to mention that angular momentum of a charged object – like an electron – is related to its magnetic moment, which is a measure of how its energy changes in a magnetic field. So the most obvious way to see the effect of spin is to look at how the energy levels of the atom shift when you an apply a magnetic field.
Long story short, the behavior of an atom doesn’t match what you’d get if you just considered the angular momentum of the electron (which can be thought of as due to the electron orbiting around the atom). Rather you have to add in a second term, which some people initially speculated was due to the electron spinning on its axis. An important physicist named Wolfgang Pauli was the first to note the need for this term, but he rightly shot down the suggestion that this was due to the electron spinning on its axis, because in order for something so small and light to have that much angular momentum from spinning, it would have to spin so fast that points on its surface move faster than the speed of light! So obviously the electron (and other atomic and subatomic particles) aren’t actually “spinning” in the classical sense, they just have an intrinsic property that shows up in our equations the same way a spin would. So we call this “spin” anyway.
Well, here we come to my “However, see below” from the first line of this post.
According to Einstein, E[sup]2[/sup] = p[sup]2[/sup]c[sup]2[/sup] + m[sup]2[/sup]c[sup]4[/sup]. (Here E is energy, p is momentum, m is mass, and c is the speed of light). You can see how for an object at rest (p = 0) we just have the better known equation E = mc[sup]2[/sup]. So we can call mc[sup]2[/sup] the “rest energy” of a body. Some text books define a “relativistic mass” which is equal to the total energy (not just rest energy) of the particle divided by c[sup]2[/sup]. This is presumably what sweet evil jesus means by saying “photons have mass.” However, in my experience most physicists prefer to use “mass” to mean “rest mass”, in which case a photon (i.e., a particle of light) is massless. Nevertheless, photons have momentum, but we have to use a more general definition of momentum than mass times velocity.
Specifically, we can treat the longer form of Einstein’s equation as a definition of momentum in terms of energy and mass. For a massless body (m=0) the equation simplifies to E = pc. So the momentum of a photon is just its energy divided by its speed.
But is defining momentum in this way consistent with our previous definition of mass times velocity? As a matter of fact, it is, provided the velocity of the particles in question is much less than the speed of light.
The total kinetic energy of an object (i.e., the energy of motion) should be the total energy minus the rest energy.
In other words:
E[sub]kinetic[/sub] = (p[sup]2[/sup]c[sup]2[/sup] + m[sup]2[/sup]c[sup]4[/sup])[sup]1/2[/sup] - mc[sup]2[/sup].
Factoring out an mc[sup]2[/sup], we have:
E[sub]kinetic[/sub] = mc[sup]2[/sup]( (p / mc)[sup]2[/sup] + 1)[sup]1/2[/sup] - 1)
where (p / mc) is much smaller than 1 since the velocity is much less than c. Using this fact, we can write the approximate equation:
E[sub]kinetic[/sub] = mc[sup]2[/sup](1 + (1/2)(p / mc)[sup]2[/sup] - 1) = p[sup]2[/sup] / 2m
(This approximation is called a Taylor series – you might be familiar with it from a calculus course.)
But E[sub]kinetic[/sub] = p[sup]2[/sup] / 2m is just our usual relationship between kinetic energy and momentum. If you don’t recognize it, replace E[sub]kinetic[/sub] with (1/2)mv[sup]2[/sup], which is the usual formula for the kinetic energy of a non-relativistic particle. Then we have (1/2)mv[sup]2[/sup] = p[sup]2[/sup] / 2m, which simplifies to p = mv. Hooray!
Close but not quite. You could define momentum as the force required to stop the object multiplied by the time over which that force is applied. (For a non-constant force, this must become the integral of force with respect to time.)
In other words, if an object has a given momentum, you could stop it by applying a large force for a short time, or by applying a small force for a long time.
E.g., suppose an object has a momentum of 10 kg m/s. Then you could bring it to rest by applying a 1 N force for 10 seconds, by applying a 5 N force for 2 seconds, or by applying a 10 N force for 1 second. (Of course, the force has to be in the opposite direction to the object’s velocity.)
This relationship between force and momentum is just a restatement of Newton’s second law. In differential form, it reads:
F = dp/dt
In integral form, this is:
Integral of F with respect to t (from time t1 to time t2) = p at time t2 minus p at time t1.
(I could write that a lot more compactly if I knew how to enter mathematical symbols on this board, but you get the idea.)
To extend what CalMeacham is saying: in free space, there is absolute translational symmetry: there’s no dependence on position. But in other regions, say inside a substance where atoms are arranged in a lattice, the periodicity of the lattice in space creates a corresponding periodicity in the momentum. In the crystal, because each atomic position is identical, moving exactly one lattice unit per second is equivalent to not moving at all!
Of course, it’s actual more complicated than that, but I wanted to describe how fundamental the association of momentum with symmetry really is.
My apologies for being pedantic, but none of the above statements make any sense, and it is critical to get the terminology correct in order to properly understand the concepts they represent. Momentum–which has units of massspeed, like kgm/s–is not a force (which has units in terms of massacceleration, like kgm/s[sup]2[/sup]) and it is not kinetic energy. I don’t know what a “potential force” is; there is potential energy, and there are “potential” fields (electrical, magnetic, gravitational) but in mechanical terms, there is either a force applied (or in the case of non-inertial frames, apparent) or there is none.
Momentum is a vector quantity of an object’s dynamic state–i.e. you need to specify the direction and coordinate frame in which you state the momentum–while kinetic energy is a scalar quantity that says nothing about where an object is going or where it came from. A particular amount of momentum will be associated with a certain kinetic energy, but there are an infinite number of different vector sums of momentum that can give the same kinetic energy. The momentum of a closed system is always conserved; the kinetic energy of any system with collisions or interactions that are not entirely conservative or elastic will change over time. “Energy methods” are often useful in dynamics (particularly orbital dynamics) when you don’t want to bother with figuring out positions and only need to know maxima and minima.
The concept of inertia is resistance to a change in velocity, i.e. making it go faster, or slower, or in a different direction, but inertia doesn’t have a measure per se, and there are no equations that use inertia as a quantity. Inertia is best thought of as a Newtonian principle that ensures that momentum is conserved, but it is not a force or energy itself, just as the second law of thermodynamics relates temperature, energy, and entropy, but is itself not a quantifiable value.
Now, there is the concept of impulse, which is the integral of force with respect to time, or in discrete terms, a force multipled by the time interval of application. A particular impulse gives a specific change in momentum, though it doesn’t tell us anything in particular either about the force or the time it is applied. [Mass or weight] specific impulseis generally used as a qualitative way of comparing the performance of jet or rocket propulsion systems, but by itself doesn’t say anything about the power efficiency or total thrust (force) developed by an engine. There are also higher order analogues of acceleration and impulse, called jerk and yank. There are allegedly higher order terms (colloquially known as grab, snatch, and shake as time derivatives of position, and snap, crackle, and pop as higher order time derivatives of force, but I’ve never seen these in engineering literature or had any call to use them in practice.)
CalMeacham and tim314 have aptly explained angular momentum and spin, and I can add nothing that would further clarify or expand upon these concepts.
Stranger
Thanks to everyone, but Stranger has raised a new question (actually a whole bunch of questions, but I will look the rest of them up). What is the difference between impulse and momentum? I notice that they are expressed in the same units. Is it kind of like specific gravity and density?
Ft = impulse = momentum = mv, correct?
Also, with respect to spin, is it the case that the observed classical angular momentum of an elementary particle system like an atom doesn’t work out unless you add in the spin of all the constituent particles? To calculate the angular momentum of the solar system, does one need to factor in the rotation of the planets and moons? How do we know that the missing factor is an intrinsic property of the particle? Is it because when we add a particular particle to a system, that system’s angular momentum is increased or decreased by the spin of the new particle? Do we know what causes spin? It sounds like a fudge factor, except that I know that it comes in discrete multiples of h-bar.
Thanks for your help,
Rob
If I apply a constant force F to an object for a time interval [symbol]D[/symbol]t, the object’s momentum will change by [symbol]D[/symbol]p=F[symbol]D[/symbol]t. This product of force and time interval (of application of that force) is called impulse. Note that the impulse does not equal the momentum. It only gives us the change in momentum.
More generally, impulse is the sum of such products. Bringing calculus in allows one to take the limit [symbol]D[/symbol]t–>0, giving the definition as an integral:
impulse = integral(F(t)dt).
Note that the relation [symbol]D[/symbol]p=F[symbol]D[/symbol]t is just the familiar F=ma when you substitute [symbol]D[/symbol]p=m[symbol]D[/symbol]v and a=[symbol]D[/symbol]v/[symbol]D[/symbol]t.
Impulse is a change in momentum, just like work is (one kind of) change in energy. You wouldn’t be too inconvenienced if you just used the one term.
For a system the size of the Solar System, the spins will be in every which way, and will mostly cancel out to something insignificant compared to the classical angular momentum. But for smaller systems like a single atom, the contributions of the different sorts of angular momentum are significant and detectable. Basically, we can tell that an atom with +1 orbital angular momentum and one kind of spin has the same total angular momentum as another atom with 0 orbital angular momentum and a different kind of spin. So we deduce that the first one has -1/2 spin and the second one has +1/2.