# Equilibrium of a Rod

burnst14
Problem Statement: As shown in the figure below, a uniform beam is supported by a cable at one end and the force of friction at the other end. The cable makes an angle of θ = 30°, the length of the beam is L = 1.50 m, the coefficient of static friction between the wall and the beam is μs = 0.420, and the weight of the beam is represented by w. Determine the minimum distance x from point A at which an additional weight 2w (twice the weight of the rod) can be hung without causing the rod to slip at point A.

Relevant Equations:
Ff = μN
Ʃτ = 0
ƩF = 0

Solution Attempt:
I'm not really sure where to start. Does Fn (normal force) come from the weight of the rod multiplied by cos(30) due to the angle of the cable holding it against the wall? And once I find Fn, I multiply that by μ, giving the upward force of friction. Then, placing the rotational axis at the end of the rod that the cable is attached to (because it's not moving, I can place the axis anywehere) and set Fn equal to the weight of the object being attached multiplied by the distance from the wall. Would that work?

voko
You just need to go about the problem systematically.

Write down the equation for x-components of the forces.

Write down the one for y-components.

Then for moments/torques.

You are going to have a bunch of known and unknown forces in those equations. Of those, the normal force at A and friction at A will be related with the coefficient of friction.

1 person
burnst14
Okay so:

ƩFy: Ff + FBsin30-2w-w = 0
ƩFx: FN - FBcos30 = 0
Ʃτ: -w(0.75) - 2wxm + FBsin30(1.5)

Is that correct?

voko
Assuming the third formula ends with " = 0 ", that looks good.

1 person
burnst14
Alright so all of that is correct, but now what? We don't have most of those variables and the only ones we can solve for are FF/FN. How do we relate w to FN? Everything has to be in terms of "w" as it stands.

What am I missing?

voko
Observer that once you relate friction and the normal force, the first to equations can be solved, giving you all the forces in the system. But you still have the third one to satisfy.

burnst14
So rewriting ƩFy we get:

μFN - 2w - w = -FBsin30 → -(μFN - 2w - w) = FBsin30

And rewriting ƩFx we get:

FN = FBsin30

We set these equal which gives:

-(μFN - 2w - w) = FN

then

-μFN + 2w + w = FN

then

(-0.420)FN + 2w + w = FN

then

What? Nothing cancels so we can't solve for anything.

voko
And rewriting ƩFx we get:

FN = FBsin30

It was FBcos 30 originally.

then

(-0.420)FN + 2w + w = FN

then

What? Nothing cancels so we can't solve for anything.

How so? Assuming it is correct (which it is not), this is the same as 3w = 1.420 FN. Surely this can be solved for FN.

Gank

burnst14
Oh yeah, so it really doesn't work then.

How so? Assuming it is correct (which it is not), this is the same as 3w = 1.420 FN. Surely this can be solved for FN.

Now I feel really dumb. How do you solve that for anything, but an answer in terms of w?

But back to the problem, I'm at a loss of what to even try. You say relate FN and μ, but I can't.

If I try setting it equal with FN rather than FB...

I get:
FBcos30 = (FBsin30 - 3w) / -μ

Does cos30/sin30 cancel somehow that I'm unaware of? That would help with one of the variables. Actually that would help everything! Waitwaitwaitwaitwait...

[FBcos30] / [FBsin30] = -3w/ -μ

substitute in for μ:

[FBcos30] / [FBsin30] = -3w/ -0.420

Cancel FB and multiply by -0.420:

-0.420(sin30/cos30) = -3w

Divide by -3:

(0.420/3)(sin30/cos30) = w

RIGHT???

burnst14

I don't have the answers. It's a Web Assign, if you'v heard of it. Basically online physics homework that allows for 5 solution attempts.

Gank
The way I did mine was:

We need to find the Fb in terms of w then we can eliminate it from the final answer as all the terms in the torques will have w. So I found 0.42Fn=Ff and that Fn=Fbcos30. From there i found Ff in terms of Fb and then put this into Fy to find Fb in terms of w. From here i substituted this into the torque equation and simplified

Gank
@voko ?

Gank
How many more answer attempts do you have?

Gank
Is there any help available on this thread?

voko
FBcos30 = (FBsin30 - 3w) / -μ

Does cos30/sin30 cancel somehow that I'm unaware of? That would help with one of the variables. Actually that would help everything! Waitwaitwaitwaitwait...

[FBcos30] / [FBsin30] = -3w/ -μ

I think you need to brush up on your algebra skills. When you have ## a F_B = (b F_B + c)/d ##, you should have no problem doing this: $$a F_B = (b/d) F_B + (c/d) \\ a F_B - (b/d) F_B = c/d \\ (a - b/d)F_B = c/d \\ F_B = \frac {c/d} {a - b/d} \\ F_B = \frac {c} {ad - b}$$

Gank

voko
The way I did mine was:

We need to find the Fb in terms of w then we can eliminate it from the final answer as all the terms in the torques will have w. So I found 0.42Fn=Ff and that Fn=Fbcos30. From there i found Ff in terms of Fb and then put this into Fy to find Fb in terms of w. From here i substituted this into the torque equation and simplified

This approach should work.

Gank
Cheers Voko. Sorry for being quite brash - I get really into problem solving

voko
The way I did mine was:

We need to find the Fb in terms of w then we can eliminate it from the final answer as all the terms in the torques will have w. So I found 0.42Fn=Ff and that Fn=Fbcos30. From there i found Ff in terms of Fb and then put this into Fy to find Fb in terms of w. From here i substituted this into the torque equation and simplified

There is also another approach, somewhat more complex but "more correct". Note that by letting ## F_f = \mu F_n ## we assume that the friction is maxed out. But we do not really know if it is going to be maxed out at a minimum or at a maximum of ##x##.

So the more correct approach would involve not letting ## F_f = \mu F_n ## just yet, and instead solve the system of three equations in three variables ##F_B, \ F_f, \ F_n ## and one parameter ##x##. Then find the range of ##x## where ## F_f \le \mu F_n ##. You could try that.

burnst14
I think you need to brush up on your algebra skills. When you have ## a F_B = (b F_B + c)/d ##, you should have no problem doing this: $$a F_B = (b/d) F_B + (c/d) \\ a F_B - (b/d) F_B = c/d \\ (a - b/d)F_B = c/d \\ F_B = \frac {c/d} {a - b/d} \\ F_B = \frac {c} {ad - b}$$

Yes I could do that, but in this case "c" is another variable, "w". So my solution would be in terms of w like I said. Wouldn't it?

voko
Yes I could do that, but in this case "c" is another variable, "w". So my solution would be in terms of w like I said. Wouldn't it?

Yes, you will end up with ##F_B## in terms of ##w##. Which is what you want, because the torque equation is also in terms of ##w##, so you could expect it will cancel out.

1 person
burnst14
OOOOOOOOOOOHHHHHHHHHHHH!!!!!

Dang it. I was so focused on solving for a number that I completely forgot about the possibility of substitution and solving it as a system. Wow. Sorry. Okay.

burnst14
Okay I'm going to bed, I'll work on this in the morning and post what I got and let you know if I got it correct. Thanks for the patience.

burnst14
If absinθ = c
does (asinθ)b = c?

burnst14
Nevermind. I just proved it does.

burnst14
Alright I got it wrong. Here's what I did:

Using that algebra step, I got:

FB = 1.171w

Then I substituted that in for FB in:

Ʃτ: -w(0.75) - 2wxm + FBsin30(1.5)

Giving:

-w(0.75) - 2wxm + (1.171w)sin30(1.5) = 0

Combine like terms and multiply out:

0.13w = 2wx

Divide over:

x = 0.065m

This is wrong according to Web Assign. It seems too small, but who am I to say so. So where did I stray?

voko
I obtain ## F_B = \frac {3w} {\mu \cos \theta + \sin \theta} \approx 3.473 w ##.

1 person
burnst14
Alright I got the right answer using your value. I'm not sure how you got it though. That seems to be new to me.

[cos30-(sin30/-μ)]FB = 3w/-μ

FB = [(-3w/-μ)/(cos30-(sin30/-μ)]

FB = (-3w/-μ)[(1/cos30)-(-μ/sin30)

FB = (3w/μ)(sin30+μcos30/sin30cos30)

FB = (3wsin30+3wμcos30)/(μsin30cos30)

Where do you go from there?

burnst14
Also what tool are you using to write your equations? It's much more clearly written.

voko
Alright I got the right answer using your value. I'm not sure how you got it though. That seems to be new to me.

[cos30-(sin30/-μ)]FB = 3w/-μ

FB = [(-3w/-μ)/(cos30-(sin30/-μ)]

This is almost exactly the equation I got. Just multiply the numerator and denominator by ##\mu##.

FB = (-3w/-μ)[(1/cos30)-(-μ/sin30)

This is wrong. You cannot go from ## \frac {a} {b + c} ## to ## a(\frac 1 b + \frac 1 c)##, these are not equal.

As for the tool, click the Quote button on my message, you will see some code there. That code is LaTeX. When you are in (advanced) reply screen, there is a ##\sum## button, which you can use to find codes for whatever math symbol you need. Or you can just search for a LaTex reference online.

1 person
burnst14
Alright thank you. Oh okay. I'll use that next time. Thanks for all the help. This forum is great.