# Static and dynamic(kinetic) coefficients of friction

1. May 18, 2012

### thestudent101

I have a Mathematics C assignment, with one question being about static and dynamic friction. But I think it fits this forum. Anyway, we need to conduct experiments to show if there is a difference between static and dynamic friction or not with three different surfaces.

The weight of the object being tested is 0.0685kg and gravity has been assumed as -9.8m/s^2

I have calculated the coefficient of static friction for the three surfaces. This was done so by placing an object of weight 0.0685kg on a surface and increasing the angle until the object starts to slide. For wood, the average angle was 37.3°, for glass - 18.0° and for metal - 14.7°. For calculating the static coefficient of friction the weight force has been calculated as
w=mg
w=0.0685*-9.8
w=-0.6713 newtons

This has then been subbed into
0=wsinθ-Fr
0=-0.6713*sin(37.3)-Fr
Fr= 0.407 newtons

0=-wcosθ+N
0=-(-0.6713*cos(37.3))+N
N=0.534 newtons

μ=Fr/N
μ=0.407/0.534
μ=0.762 (wood)
This process has been repeated for glass (μ=0.324) and for metal (μ=0.263)

Now I am completely stuck on how to calculate the coefficient of dynamic friction. I was thinking about timing how long the object takes to slide down a certain length from a set angle? But I don't know how to calculate the coefficient from that? Any ideas and help will be appreciated thanks.

2. May 18, 2012

### PhanthomJay

You might note from your correct calculations that the static coefficient of friction is simply u_s = tan theta, which you can deduce using letter variables instead of plugging in numeric results. Does this give you a hint as to how you might calculate the dynamic coefficient of friction without using a clock or meter stick?

3. May 18, 2012

### thestudent101

The friction and normal force also need to be calculated so that's why I did it that way. But no, unfortunately that didn't give me any hints.

4. May 18, 2012

### PhanthomJay

How about if the object slides down the incline at constant velocity (no acceleration)?

5. May 18, 2012

### thestudent101

i still don't know how to incorporate that into it.

6. May 18, 2012

### PhanthomJay

I am not familiar with what is covered in Maths C. You could certainly do it the way you suggested, that is,
where knowing the time, length, and angle, you can calculate the acceleration, then use newton's second law, F_net =ma, to solve ultimately for the kinetic friction coefficient. If you are familiar with this method, this will work fine, with the usual error in the time measurement.

On the other hand, you seem to be familiar with newton's 1st law, which says the net force is zero when an object is not moving (or just about to move). That's how you calculated the coef of static friction (u_s =tan theta). Newton's first law also says that the net force is 0 if an object is moving at constant velocity. So the same equations apply, only this time you must adjust the angle such that the object is moving at constant speed without accelerating. u_k is always equal to or less than u_s, almost always less. Can you think of a way to find this angle such that u_k = tan theta when the object is moving at constant speed ??

7. May 18, 2012

### thestudent101

what's u_s and u_k?
that's probably a really stupid question.

8. May 18, 2012

### Whovian

u_s is coefficient of static friction, while u_k is coefficient of kinetic friction.

9. May 18, 2012

### PhanthomJay

Oh, sorry, u_s is the coefficient of static friction, and u_k is the coefficient of kinetic (dynamic) friction. I should have written it in LateX $\mu_s$ and $\mu_k$

10. May 18, 2012

### thestudent101

T+Wsin(theta)-Fr+n-Wcos(theta)=ma
T+Wsin(theta)-Fr+n-Wcos(theta)=0
Therefore -Wcos(theta) and N are the same.
T+Wsin(theta)-Fr=0
I'm soooo confused now.

11. May 19, 2012

### PhanthomJay

You need to look along the incline direction only when the object is accelerating; it is not accelerating perpendicular to the incline, so you still have N = mgcostheta in that direction. And what is T? No such force exists. Your first equation should read Wsintheta - Fr = ma. Where Fr = u_k(N).

But why go through all the maths when you can simply start at a small angle and give the object a nudge so it starts sliding. it will quickly stop. So then increase the angle and nudge it again. See if it still stops. keep on increasing the angle until it doesn't stop. Play around with it. When it neither stops or accelerates, that's the angle you want. And you don't have to fool with newton 2, because it's moving at constant speed, so you have, at that angle, Wsintheta -Fr = 0, as before.

12. May 19, 2012

### thestudent101

Ok, so I did what you suggested. I used the formula wsintheta-Fr=ma and n=wcostheta.
Here's the results.
Wood (static): 0.762
Wood (kinetic): 0.345
Glass (static): 0.324
Glass (kinetic): 0.212
Metal (static): 0.263
Metal (kinetic): 0.140

Do these values seem reasonable?

13. May 19, 2012

### PhanthomJay

The friction coefficients depend on the material of the 2 contact surfaces, the cleanliness of the surface, whether it is dry or moist, etc. . You do not indicate what material the obect on the incline is made of, so I don't know if your calculated coeffficients, both static and dynamic, are reasonable. Could be a lot of lab errors. Compare your results to published tables for these values.

Try your experiment again. And again. The tangent of the angle at which it just starts to move is the static coefficient friction. The tangent of the angle at which it slides at constant speed after being 'nudged', is the kinetic coefficient. Start with small angles and slowly increase the angle until these events occur.