Solving Friction: Acceleration and Velocity

[primarygun]

Imagine there is an object on the surface.
The surface is uniform smooth(with same friction over the all surface and friction is greater than 0).
Let's define the friction be f.
Act a f force to horizontally to the object continuously, it is continuous added.
It is obviously stationary.
I'm quite confused of here then. If there is 1N force pushed to it in the same direction to the f force added by hand.
What will be observed?
a.Move with acceleration first, then remains the velocity later on.
b.1.If f>kinetic friction, then the object accelerates first then decelerates to stop.
b.2. If f> kinetic friction, then the object will accelerates continuously forward.
Which one is correct?

 

[maverick280857]

Okay I think I see what you mean but you've not worded it correctly. First off, the friction coefficient is defined differently for zero and nonzero relative motion. In the former case, it is called the coefficient of static friction $$\mu_{s}$$ and in the latter, the coefficient of kinetic friction $$\mu_{k}$$. Usually $$\mu_{k} < \mu_{s}$$.

For the simple case described by you, the actual static frictional force f and the maximum possible static frictional force are related by the inequality,

$$f \leq f_{s,max} = \mu_{s}N$$

where N (=mg for zero vertical motion of the block) is the normal reaction on the block.

You need to know that for as long as f is less than fsmax, friction is a readjusting force..it equals itself to the applied force so that the net force on the body is zero and there is zero relative motion. Now if F is the applied force which is gradually increased from 0 to f, no motion will occur. At F = f, motion is "just" about to occur. As soon as F > f however, motion occurs but now f equals $$f_{k} = \mu_{k}N$$ which is less than its original value (and therefore also less than F). Hence there is a net force on the body.

This explanation can be made clearer if you draw a graph of the friction force versus applied force. At F = fsmax, there is a kink in the graph (which consists of a straight line of slope = 1 for F < fsmax and a horizontal line for F > fsmax) which corresponds to this dynamic switch from static to kinetic friction (in mathematical terminology the frictional force is not a continuous function of applied force). This graph of course can be obtained through experiment but since it holds for most simple pairs of surfaces you can find it in your physics textbook.

Now attempt your question again. Hope that helps...

Cheers
Vivek

 

[primarygun]

b2? right?

 

[maverick280857]

Yup. Good luck! :-)

 

[primarygun]

How can we prove the presence of kinetic friction?

 

[maverick280857]

Well its not something that can be proved mathematically but it can be proved logically using an argument such as the one that follows:

Suppose we exert a force F on an object that is resting on a surface whose coefficient of static friction is known (perhaps through tables or a measurement carried out using more advanced methods than the simple theoretical ones). As mentioned earlier the body will not move for as long as F is less than the maximum static frictional force. Now if F is gradually increased to a point where it "just" exceeds the maximum static frictional force, motion "just" begins and if the force F is kept constant in magnitude and direction, the acceleration vector of the body is constant (assuming no nonlinear behavior of the friction force). The net acceleration of the body is given by

$$a_{net} = \frac{F_{net}}{M}$$

and the acceleration that the applied force F would alone impart the body is

$$a_{F} = \frac{F}{M}$$

If $$a_{F} < a_{net}$$ (as will be the case) there must be some force which reduces the effect of F and that force can be safely thought of as the kinetic frictional force. Upon subsequent measurements you will find that it is less (usually) than the static frictional force.

Again, this "argument" depends on a LOT of assumptions some of which are implicit and so cannot be taken as an experimental or theoretical proof of friction, which is already known to exist. What you can prove however, through this approach is that kinetic friction exists after motion starts and not before it. And taking this a bit further you can plot the graph I had mentioned earlier.

Hope that helps...

Cheers
Vivek