Will an object ever stop because of friction
If an object is experiencing some form of resistance, such as friction or air resistance, will this force ever cause it to stop completely. For example when a car does and emergency stop does it in fact stop?
The obvious answer is yes but it would appear that it doesn’t because of the following:
1) The resistive force is some function of the objects velocity and is probably either proportional to its velocity or the square of its velocity.
2)When an object is at rest it experiences no resistive force, if it did it would start moving again.
This means that as a objects velocity approaches 0 the force that is causing it to slow down is also approaching 0. This means that the velocity of the object wont actually reach 0 but will become infinitely small.
You may therefore think that there is some sort of minimum friction, in addition to the component which is proportional to the objects velocity but this would mean that there would still be friction when the object is at rest so the object would then begin to move backwards.
Any comments or further thoughts would be appreciated to help explain this.
By David Woodford
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Hi! This is actually a very interesting question — perhaps a modern version of Zeno’s paradox. Let me try to put it into mathematical terms.
Suppose we have an object moving along the x axis with constant velocity v0. At time t=0, the object is at x=0 and it begins to be subject to a friction — a force with direction opposite to its movement and which is a function of its velocity.
First, let us suppose friction is directly proportional to the object’s velocity. Then the acceleration of the object from t=0 onwards is
a = -Cv,
where C is some positive real constant (the sign ensures friction opposes to movement). Since the acceleration is the first derivative of the velocity v, let us say a = v’ and write:
dv/dt = -Cv.
A differential equation! Now comes the hard math. There is a theorem for differential equations of the form
dv/dt = P(t) v,
where P(t) is some continuous function, which states the solution in t for initial condition v(t=0) = v0 is
v(t) = v0 e^(-A(t)), (1)
where A(t) is the integral of P(x) from 0 to t.
So in this case, since P(x) = -C, we get A(t) = -Ct and the solution is
v(t) = v0 e^(-Ct).
The conclusion is obvious. No matter how big C is, since the exponential never drops to zero, the object will never stop, even though its velocity will get as close to zero as you want, if you wait enough. Your intuition is correct.
We could try to get around this by saying: “oh, but friction in fact is not proportional to v, that’s just an approximation. If we plug in something other for P(t) above, perhaps we can make v equal zero after a certain amount of time”. Well, you see, the problem is that, as long as P(t) is continuous, the solution is still in the form (1) above, and we still have the problem that the exponential is never zero.
So what do we do? An easy way out is not to take mathematics so seriously. After all, a really small number is practically zero, so we don’t have to worry, right? But this is not a comfortable answer, because we chose mathematics to be the language of Nature and it leaves us wondering how many other “wrong” mathematical facts are out there.
I do not know what the correct “way out” is, although I’m fairly sure someone has tackled the problem before. But because you ask for plausible explanations, I’ll give you two possible “ways out”:
1) Friction is usually understood as an “average” force. For instance, when atmospheric drag slows down your car, that drag is in fact made up of billions of tiny collisions between molecules moving about erratically and your car. We arrive at the C value I used above by making some assumptions about the distribution of those random events and taking a mean. But on a small scale of time, the actual behavior might deviate from the mean — you might have some small “gusts” of wind here and there yielding a net force that is not proportional to the velocity. This may have consequences when the object has slowed down enough and is more sensitive to these forces.
2) The other explanation makes an appeal to Quantum Mechanics. We now know Nature is not “continuous” as we though it was a century ago. In other words, certain physical quantities cannot take any value in an interval — there is only a discrete number of possibilities and the quantity “jumps” from one value to another never passing through any value in between. One of these quantities is velocity (momentum, to be precise). So it may be the case that the solution of the differential equation is wrong for small values of v because we make the assumption that P(x) is continuous, when in fact it isn’t.
Im doing A level maths and further maths at the moment with the hope of starting university in October (ill be 18 in july). I live in the UK but im not sure how that converts to american stages of education(Im guessing your american simply on the basis that the majority of people on the internet are).
If your interested below is one of last years pure maths papers so you can see the level of covered ive covered.
http://www.wjec.co.uk/uploads/papers/s08-979-01.pdf
Dave
Terrific web site=) Hope to come back again…
Super writing=D hope to visit once again..
It seems to me that the response from T is overcomplicating the problem. Frictional forces are in general not proportional to velocity, in fact the static frictional force (the force required to get a static object moving over a rough surface) is usually greater than the dynamic frictional force (the force required to keep it moving at a constant speed).
Thanks for your response,
Frictional forces may not be proportional to the velocity but they clearly depend upon it a increase and as the velocity increases. Does static friction have an effect in this situation since there isn’t a force being applied to the object to start it moving?
My main point in this post is that when no force is being applied to an object and it is at rest the frictional force is 0 so as a moving objects velocity is reduced and tends to 0 the frictional force that it is experiencing is also reduced and also tends to 0 so it would appear the fictional force could never bring the object to rest.
thanks again for your comment,
David
Doug:
Because we are talking about the effect of friction over a *moving* object, static friction is irrelevant in this situation.
Dynamic friction clearly depends on velocity, and that’s about all you can say about it in a general way (that and the fact that it opposes movement). Postulating that friction is directly proportional to velocity is a simplification — one frequently used in physics problems. I only did so in order to be able to arrive at a concrete result. But, as I pointed out, my conclusion generalizes to cases where the dependence is something other than a linear function.
I’m not sure what you mean when you say that the dynamic frictional force is required to keep an object moving at a constant speed. In fact an object will move at a constant speed if there is no force at all (or if the resultant of all forces is zero).
Thanks for your detailed response,
Both of your explanations seem to offer a good explanation so thank you for the response.
David
No problem! Meanwhile I re-read my reply and noticed a couple of errors (an extra minus sign in (1), v’ instead of dv/dt, P(x) instead of P(t)), but I think they were fairly easy to spot.
If I may ask, what is your level of education? From the title of your blog it seems you are in high school, but you do cover slightly more advanced stuff.
I’m actually from Portugal.
I’m 22 and about to finish a Master’s degree in Electrical Engineering. Besides that I have a strong passion for mathematics and computer programming.
You do cover advanced stuff in comparison with us here in Portugal; I wasn’t taught anything about integrals before University (much to my dismay). We really should have our curricula revised.
Best of luck with your studies!