# At what angle of force is Kinetic friction least?

1. Mar 25, 2015

### gijungkim

1. The problem statement, all variables and given/known data
The total force needed to drag a box at constant speed across a surface with coefficient of kinetic friction Uk is least when the force is applied at an angle theta such that

a.sin(theta) = Uk.

b.cos(theta) = Uk.

c.tan(theta) = Uk.

d.cot(theta) = Uk.

e.sec(theta) = Uk.

2. Relevant equations
F = ma
Ff = Uk * Fn = m*g*Uk
Fx = F *cos(theta)
Fy = F*sin(theta)

3. The attempt at a solution
First the answer is d. cot(theta) = Uk
I tried to solve it, but couldn't figure it out...
This is my work
Since it is dragged at constant speed, the acceleration is 0. So a = Ftotal/m = 0
Ftotal = F*cos(theta) - (mg - F*sin(theta))Uk = 0
F*cos(theta) / (mg - F*sin(theta)) = Uk... which is wrong...
Can anyone help me?

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2. Mar 25, 2015

### Staff: Mentor

So far, so good.

Realize that F is a function of theta. You'll need a bit of calculus to figure out the value of theta that minimizes F.

Question: Did they give a diagram defining the angle theta?

3. Mar 25, 2015

### gijungkim

They did not give me any diagram defining the angle theta.. Can you help me out little bit more using calculus?? I still don't get it :(

4. Mar 25, 2015

### Staff: Mentor

How would you use calculus to find the max or min of a function?

5. Mar 25, 2015

### gijungkim

You derive the function and make it as a 0 ??

6. Mar 25, 2015

### Staff: Mentor

Yes! You'll set the derivative ($dF/d\theta$) equal to zero and solve for $\theta$.

Hint: What you have will require implicit differentiation.

7. Mar 25, 2015

### gijungkim

I need one more hint! What equation do I have to set the derivative??
F*cos(theta) / (mg - F*sin(theta)) = Uk
This one??

8. Mar 25, 2015

### gijungkim

Ftotal = F*cos(theta) - (mg - F*sin(theta))Uk = 0
Oh nevermind it will be this one right?

9. Mar 25, 2015

### AlephNumbers

I don't think this requires calculus. Just some algebraic manipulation.

10. Mar 25, 2015

### gijungkim

Ok... I tried my best... and got tan(theta)..... Ughhhh why is this so hard to me

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11. Mar 25, 2015

### gijungkim

Can you help me out???

12. Mar 25, 2015

### AlephNumbers

Were you assigned this problem in a calculus based physics class?

13. Mar 25, 2015

### gijungkim

Nope. I'm studying physics by myself so I can take AP physics on May! I don't have anyone to ask questions beside here and I got this questions from internet!

14. Mar 25, 2015

### AlephNumbers

Try solving for the coefficient of kinetic friction in terms of F and itself. You can then substitute this value ...somewhere...

15. Mar 25, 2015

### AlephNumbers

16. Mar 25, 2015

### Staff: Mentor

Given how you defined theta, I would say that was correct. (That's why I asked if you had been given a diagram.)

17. Mar 25, 2015

### gijungkim

Ahh physics is killing me hahaha

18. Mar 25, 2015

### gijungkim

So when do I get cot(theta)?? where does the theta have to be to get an answer cot(theta)??

19. Mar 25, 2015

### Staff: Mentor

Since cot(theta) = 1/tan(theta), you'd get cot(theta) if you let theta = angle with the vertical.

Where did you find the problem?

20. Mar 25, 2015

### gijungkim

I found this question from internet! I think this question is based on the book "Physics for scientist and engineers" Serway and Jeweet, but I already read the book and couldn't find anything that explains this problem haha