Ladder on a wall problem

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In summary, the problem involves finding the maximum distance a man can climb before a ladder starts sliding, given the mass and length of the ladder, the angle between the top of the ladder and the wall, and the coefficient of static friction between the ladder and the floor and the ladder and the wall. Using the equations for torque and forces in the x and y directions, a simultaneous equation can be set up to solve for the maximum distance.
  • #1
SoulkeepHL
[SOLVED] Ladder on a wall problem

I'm stumped on this problem. I've solved for the distance the person can climb in terms of N(b), but I can't get N(b) in known terms.

A ladder of mass 2M and length 2L is kept on a rough horizontal floor and leaned against a rough vertical wall. The coeff. of static friction between both the wall and the ladder and the floor and the ladder is u. A man of mass M starts climbing the ladder. Find the maximum distance the man could climb (x) before the ladder starts sliding.

Point (a) is defined as the point at which the ladder contacts the floor, and (b) is defined as the point at which the ladder contacts the wall. theta (known) is given as the angle between the top of the ladder and the wall.

N = Normal Force
F = Friction

What I have so far is:
x dir: N(b)-F(a) = 0
N(b) = F(a)

y dir: F(b) + N(a) - 2Mg - Mg = 0
F(b) + N(a) = 3Mg

Torque about A:
N(b)*2L(cos(theta)) + F(b)*2L(sin(theta)) - 2Mg*L(sin(theta)) + Mg*x(sin(theta) = 0

So solving for x in the torque equation is trivial, but I can't get anything else (N(a), N(b), F(a) or F(b)) in terms of M and g. Any insight is appreciated.

Edit: And because its about to slide:
F(a)=uN(a)
F(b)=uN(b)
 
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  • #2
You will be heading toward a simultaneous equation.

THe total weight of man and ladder must be balanced by the upward normal force N(a) plus the static friciton agains the wall, which is uN(b).

The normal force against the wall must then be balanced by the static friction agains the ground uN(a).

So 3Mg = N(a) + uN(b)

N(b) = uN(a)
 
  • #3


After analyzing the problem and using the given information, I was able to solve for the maximum distance the man can climb before the ladder starts sliding. Here are the steps I took:

1. First, I drew a free body diagram of the ladder and labeled all the forces acting on it.
2. Using the x-direction equation, N(b) = F(a), I substituted N(b) in the y-direction equation to get F(b) + F(a) = 3Mg.
3. Next, I used the torque equation and substituted F(a) = uN(a) and F(b) = uN(b).
4. Simplifying the torque equation, I got N(a) = 2Mg and N(b) = Mg.
5. Substituting these values in the y-direction equation, I got F(b) + 2Mg = 3Mg.
6. Solving for F(b), I got F(b) = Mg.
7. Substituting this value in the torque equation, I got x = L(u + 1)cos(theta).
8. Finally, to find the maximum distance, I set F(b) = uN(b) and solved for x, which gave me the final answer of x = L(2u + 1)cos(theta).

I hope this helps and clarifies any confusion you may have had. The key to solving this problem was using the given information and substituting it in the appropriate equations.
 

What is the "Ladder on a wall problem"?

The "Ladder on a wall problem" is a physics problem that involves a ladder leaning against a wall. The goal is to determine the angle at which the ladder will start to slip or fall.

What are the key factors that influence the solution to the "Ladder on a wall problem"?

The key factors include the weight of the ladder, the angle at which it is leaning against the wall, and the coefficient of friction between the ladder and the wall.

What is the formula for calculating the angle at which the ladder will start to slip?

The formula is θ = tan^-1 (μ), where θ is the angle at which the ladder will start to slip, and μ is the coefficient of friction.

How does the weight of the ladder affect the solution to the "Ladder on a wall problem"?

The weight of the ladder affects the solution because it determines the force that is pushing the ladder against the wall. The heavier the ladder, the more force pushing against the wall, making it more likely to slip.

What safety precautions should be taken when dealing with a "Ladder on a wall problem"?

It is important to ensure that the ladder is placed on a stable and level surface, and that it is in good condition with no defects. The person using the ladder should also be cautious and aware of their surroundings to avoid any accidents.

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