At what angle of force is Kinetic friction least?

[gijungkim]

Homework Statement

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.

Homework Equations

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

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?

 

[Doc Al]

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

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?

 

[gijungkim]

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?

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 :(

 

[Doc Al]

Can you help me out little bit more using calculus??

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

 

[gijungkim]

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

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

 

[Doc Al]

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

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

Hint: What you have will require implicit differentiation.

 

[gijungkim]

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

Hint: What you have will require implicit differentiation.

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

 

[gijungkim]

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

Hint: What you have will require implicit differentiation.

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

 

[AlephNumbers]

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

 

[gijungkim]

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

Hint: What you have will require implicit differentiation.

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

 

[gijungkim]

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

Can you help me out?

 

[AlephNumbers]

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

 

[gijungkim]

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

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!

 

[AlephNumbers]

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

 

[AlephNumbers]

Also, I believe you made a sign error in your forces.

 

[Doc Al]

Ok... I tried my best... and got tan(theta)...

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

 

[gijungkim]

Also, I believe you made a sign error in your forces.

Ahh physics is killing me hahaha

 

[gijungkim]

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

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

 

[Doc Al]

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

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?

 

[gijungkim]

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?

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

 

[gijungkim]

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?

Can I ask you one irrelevant question? Do you know where I can find good Physics Mechanics problem? Since I'm studying by myself, I just like to read the book and solve problems from online and ask questions here if I get stuck. I found good problems from the book I mentioned above, but I'm almost done with that so I'm just wondering if you knew any or if you have any haha

 

[Doc Al]

Can I ask you one irrelevant question? Do you know where I can find good Physics Mechanics problem? Since I'm studying by myself, I just like to read the book and solve problems from online and ask questions here if I get stuck. I found good problems from the book I mentioned above, but I'm almost done with that so I'm just wondering if you knew any or if you have any haha

There are many books of problems available, at various levels. To start off, you might want to try something like https://www.amazon.com/dp/0071763465/?tag=pfamazon01-20.

 

[gijungkim]

There are many books of problems available, at various levels. To start off, you might want to try something like https://www.amazon.com/dp/0071763465/?tag=pfamazon01-20.

Thank you!