A simple ladder problem (equilibrium)

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SUMMARY

The discussion centers on solving a ladder equilibrium problem involving a uniform ladder of 5.0 m resting against a frictionless wall. The ladder weighs 160 N, and a man weighing 740 N climbs it, with a static friction coefficient of 0.40 at the base. Key calculations include determining the maximum frictional force (360 N), the actual frictional force when the man climbs 1.0 m (171 N), and the maximum distance the man can climb before the ladder slips (2.7 m). The use of trigonometric functions is critical for accurate calculations, specifically recognizing the ladder forms a 3-4-5 triangle.

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  • Understanding of static equilibrium principles
  • Knowledge of free-body diagrams
  • Familiarity with trigonometric functions in physics
  • Experience with frictional force calculations
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  • Review static equilibrium problems involving ladders and forces
  • Learn about free-body diagram construction for complex systems
  • Study the application of trigonometric ratios in physics problems
  • Explore the effects of varying coefficients of friction on stability
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NutriGrainKiller
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First test on equilibrium tomorrow, and I'm very sure there is going to be a ladder-type problem on it where I have to determine normal forces, frictional forces when someone is on the ladder up a certain distance, and how far can the person go up before it starts to slip. I don't know if you guys are familiar with masteringphysics.com or not, but it is an online homework system where it automates questions similar to that in the book. I've gotten pretty far in the problem, but for some reason I can't complete the 2nd two problems. Here is the exact problem on mastering physics:
A uniform ladder 5.0 m long rests against a frictionless, vertical wall with its lower end 3.0 m from the wall. The ladder weighs 160 N. The coefficient of static friction between the foot of the ladder and the ground is 0.40. A man weighing 740 N climbs slowly up the ladder.
A)What is the maximum frictional force that the ground can exert on the ladder at its lower end?
B)What is the actual frictional force when the man has climbed 1.0 m along the ladder?
C)How far along the ladder can the man climb before the ladder starts to slip?
I drew a diagram and a free-body diagram of the problem, and was able to get the answer to A through the requirements for equalibrium. Here is my work:
(Fnet)x: Fs-Fw=0
(Fnet)y: Fn-Fp-Fg=0
We have Fp and Fg, so Fn = Fp+Fg or Fn = 900
Since Fs <= Us*Fn we have (Fnet)x: Us*Fn-Fw=0, or Fw=(.4)*(900), so Fw = 360. And since Fs<=Us*Fn, Fs = Fw, so Fsmax=360. This is the answer to A.
I am confused as hell when it comes to B. It is difficult to explain because you cannot see my free-body diagram, but I will do my best to describe it.
Don't know how to use tao, so I'll just use lowercase t.
(tnet)=tn+ts+tp+tg+tw=0. I am choosing the origin to be at the lower end of the ladder where Fn and Fs are, so those would equal zero because they have no moment arm. So here is the new equation:
(tnet): -(Fp)(R)Sin()-(Fg)(R)Sin()+(Fw)(R)Sin()=0. With the values we have
(tnet): -(740n)(1m)sin(53.13)-(160n)(2.5m)sin(53.13)+(Fw)(5m)sin(36.87)=0.
I found 53.13 and 36.87 through basic trig. Here are my questions: I am solving for a new Fw (normal force coming from the wall) in this equation right? i.e. I would NOT be using the value I obtained earlier. After I get that I plug it back into (Fnet)X to get Fs, which would be my answer - right?

My final answer was 301.3, which is NOT right - The correct answer is 171n. Why :confused:

When it comes to part C I'm not sure how to go about it because I don't fully understand the answer to part B. I'm guessing you solve for the unknown moment arm in the tnet equilibrium equation.

Thanks guys

pink floyd says hi :cool:
 
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You've got your angles the wrong way round!

Or, you should be taking cos instead of sin, and vice versa.

The distance from the bottom of the ladder to the wall is 3 m.
3m is not the height up the wall to the top of the ladder
 
Oh, btw, it's a 3-4-5 triangle, with cos@ = 3/5, sin@ = 4/5. It makes calculations a bit simpler if you recognise that :smile:
 
Thanks Fermat :)

btw for searching purposes the answer to B is 171n and C is 2.7m.
 
Yeah, I got those answers too.
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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