Help Solving 2 Qs: Tan(Ɵ) & Sign Direction Error

AI Thread Summary
The discussion revolves around solving two questions related to the equations of motion and friction in a system involving forces and moments. The user initially derives the inequality for tan(Ɵ) but encounters sign errors in their calculations, leading to confusion over the correct relationships. They clarify that the friction force must be correctly related to the normal force to avoid slipping, which resolves one of their issues. The user also questions the validity of their moment calculations and whether a discrepancy in the answers is due to a typo in the textbook. Ultimately, they receive confirmation that their analysis is correct, resolving their confusion.
gnits
Messages
137
Reaction score
46
Homework Statement
To find possible angle of two leaning rods
Relevant Equations
moments
Could I please ask for help with the following question:

1642514927301.png


The last part follows easily from the first part.

Answer from back of book for first part is:

2/(3u') <= tan(Ɵ) <= 2u

What I have done is the following:

Here's my diagram (I have separated the components to show the internal forces in the system. I have used t instead of Ɵ) :

leaning.png


Orange forces are internal forces.

If I equate forces for the whole system vertically I get:

R = (3/2)W

If I equate forces vertically for the vertical rod only I get:

F1 = W/2

If I take clockwise moments about C for rod CD only I get:

F * L sin(t) + W * (L/2) cos(t) = R * L cos(t)

Substituting for R and rearranging gives: F = W / tan(t)

Now for no slipping at D we need F <= u' R so this gives:

W/tan(t) <= u' * (3/2)W

Which leads to tan(t) <= 2/(3u') which is the answer asked for but with the sign reversed.

1) How have I gotten the sign mixed up?

Taking clockwise moments about D for rod CD only I get:

S * L sin(t) = F1 * L cos(t) + W * (L/2) * cos*(t)

Substituting for F1and rearranging gives:

S = W/tan(t) and for no slipping at C we need S <= u * F1

So gives W/tan(t) <= u * W/2 which leads to tan(t) >= 2/u

Again wrong direction of sign and also wrong answer as we need tan(t) <= 2u

2) Where have a erred?

Thanks for any help.
 
Physics news on Phys.org
Thank you very much kuruman for your reply.

So, notwithstanding the signs, would you (or others) think my analysis of taking moments for CD only about D looks correct? I ask because here I differ in the actual form of the answer. The given answer is 2u but I arrive at 2/u. If you\others agree that 2/u is correct then I would put it down to a typo in the book.

Thanks again.
 
Last edited:
gnits said:
Now for no slipping at D we need F <= u' R so this gives:

W/tan(t) <= u' * (3/2)W

Which leads to tan(t) <= 2/(3u') which is the answer asked for but with the sign reversed.
I agree with the first equation here, but that implies that ##\mu' \tan \theta \ge \dfrac 2 3##.
 
PeroK said:
I agree with the first equation here, but that implies that ##\mu' \tan \theta \ge \dfrac 2 3##.
Yes indeed, I agree. I had fallen at the last there. Thanks for that. So only the second part is now causing me issues.
 
gnits said:
S = W/tan(t) and for no slipping at C we need S <= u * F1
For no slipping we need ##\mu S \ge F_1 = \dfrac{mg}{2}##.

##S## is the normal force and ##F_1## is the required friction force.
 
PeroK said:
For no slipping we need ##\mu S \ge F_1 = \dfrac{mg}{2}##.

##S## is the normal force and ##F_1## is the required friction force.
Thanks very much indeed. I had swapped S and F1. I had written S <= u * F1 whereas, as you say, I should have written F1 <= u S. And indeed this leads to the book's answer. Thank you very much.
 
gnits said:
Thank you very much kuruman for your reply.

So, notwithstanding the signs, would you (or others) think my analysis of taking moments for CD only about D looks correct? I ask because here I differ in the actual form of the answer. The given answer is 2u but I arrive at 2/u. If you\others agree that 2/u is correct then I would put it down to a typo in the book.

Thanks again.
Sorry, I deleted my reply to write a better one, but then I had an emergency that kept me away from the task. You must have read my post soon after I posted it but before I deleted it. Anyway, it looks like your issue has been resolved so there is nothing more for me to say. I apologize for the confusion.
 
Back
Top