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The Ladder Problem

A beam leaning on a wall, or a person standing on a leaning ladder is often used as an introduction to "rigid body" statics problems. The main difference between these and previous point mass problems (like block on a slope) is the addition of a 3rd equation when solving the sum of forces: torque or moment*. The other difference is the location of vectors on the free body diagram becomes important.

The Free body diagram looks something like this, keep in mind the reference angles used, a common variation measures the angle at the top. A person might stand at some distance along the length of the ladder (NOT height), if your problem doesn't have this, set m_p=0 kg. 

*these are basically synonyms, Wikipedia has a good explanation


Note that the ladder touches another object at 2 locations, so there are 2 copies of the normal and friction forces, noted with A(circle) and B(circle). Using top and bottom, 1 and 2, or anything else is fine, but if there are less than 5 points, I personally prefer the alphabet for subscripts. I chose the direction of the friction to point opposite the direction it would slide; test it with a pencil and your hand.

The sum of forces in the x and y axes are relatively simple** to sum using the same process as the block on a slope, but if the sum of moments is a new concept, read this page first.

The 1st step is to choose a point to sum the moments around, a rule of thumb is to find a point where the most lines of actions of unknown forces pass through with as few knowns as possible. Here, our options are points A, B, and the center of mass(COM). The weight is the only known force, so the  COM is not a good choice for this problem as the equation would have 4 unknowns, making the math harder. A and B have the same number of unknowns, but B also has the reference angle so it's more convenient.

Next, the magnitude of each moment can be found by multiplying the magnitude of the force by the length of the moment arm, or how far away the line of action is from the point (NOT the distance to the applied force). See just below the equation. Note that the weight acts at the COM of the ladder located at the halfway point.

Then, the sign is found with the right-hand rule. Or, If a pin was put through the point, which way would this force pull it? CCW is a positive moment, CW for negative.

**assuming it isn't moving and the wall is verticle, it gets spicy if it's not.


Now is a good time to re-read your question; right now, we have 3 equations with 4 or 5 unknowns "f_A", "f_B", "N_A", "N_B", and "d".

But newtons laws only gave 3 equations, so the question statement either eliminates a variable, such as no person/no friction at point A, or the question allows the use of another equation like maximum friction force or gives one of the values like "d" as a given. Unknown theta is easy, but I don't want to go on a Tangent.


The version we will continue with is: no friction at point A (f_A = 0), maximum static friction at point B (f_B = mu*N_B) solve for the maximum height the person can stand.


The big boxed answer is technically not correct, as my question asked for height in big bold letters not the distance up the ladder, but the one extra step to find "h" is a common trick question. For written tests, it doesn't hurt to put down both with labels.

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