Friction problem- possible error in Halliday/Resnick

[Syrus]

Homework Statement

See attachment. I am only concerned with part b). Part a) is solved.

Homework Equations

The Attempt at a Solution

See attachment. The solution is quoted to be tan^-1(μ_s) = θ₀, but, as can be seen by my solution, this is impossible based on the derivation. Is this their error, or mine?

 

[frogjg2003]

What did you do for part a?
Not sure if it will work, but try taking your answer for part a, and having v->0.

 

[Syrus]

I solved for F in part a). I display the results (verified correct) at the top of the page. I'd like to see how the value for theta naught is explicity obtained.

 

[frogjg2003]

You said the solution is tan(μ_s) = θ₀. Do you mean tan^-1(μ_s) = θ₀?

 

[Syrus]

Yes, apologies- it's edited now.

 

[frogjg2003]

Well, just take the limit when force goes to infinity. When the angle is less than θ₀, no amount of force will be able to move the mop.

 

[Syrus]

I considered that, but I wondered if there were any better ways of showing this. Plus, this example is from an elementary (Halliday/Resnick) text, and such un-straightforward proofs seem unwarranted in problem-solving. Just some thoughts. Is an explicit derivation out of the question?

 

[frogjg2003]

This seems like one of the more "challenging" problems. As such, more out-of-the-box thinking is required. This method is pretty good as it stands. What would you consider a "better" way?
Why is this an "un-straightforward proof" or not "an explicit derivation"?

 

[Syrus]

Fair enough. The only reason I said what I did above was because other problems and results derived throughout the book are a bit more explicit in using Newton's laws to demonstrate the results. I can accept this method, just inquiring about other manners.

 

[Syrus]

The value for F should be multiplied by μ_k.

 

[ehild]

You overcomplicate the problem a bit.
You got $$F=\frac{mg\mu_k}{\sin(\theta)-\mu_k \cos(\theta)}$$
The mop is pushed, so it can not be negative. What does it mean for theta?

ehild