A stepladder of negligible weight is constructed as shown in Figure where x = 1.

Question

A stepladder of negligible weight is constructed as shown in Figure where x = 1.60 m. A painter of mass 69.0 kg stands on the ladder 3.00 m from the bottom. Assuming the floor is frictionless, find the following. (Suggestion: Treat the ladder as a single object, but also each half of the ladder separately.) the tension in the horizontal bar connecting the two halves of the ladder the normal forces at A and B (at A) (at B) the components of the reaction force at the single hinge C that the left half of the ladder exerts on the right half (rightward component) (upward component)

Explanation / Answer

Hi,

In this case we have an equilibrium problem, as such the sum of forces and torques should be zero

If we consider the whole ladder as a single object we have that the balance of forces is equal to:

x axis:   0 = 0 (there are no forces in this direction)

y axis: NA + NB = Mg

If we treat each half of the ladder separately we have the following

Left haft Right haft

x axis:   T = R1x x axis: T = R2x

y axis:   NA = R1y   y axis: NB = R2y

torques: T(1 m)sin = NA(3 m)cos      torques: T(2 m)sin() = NB(4 m)cos()

The angles and is equal to:

cos = 0.8 m/3 m ::::::: = 74.5° cos = 0.8 m/4 m :::::::::: = 78.5°

So, we have seven equations with seven unknowns, therefore, when we solve all of them we have the following:

(a) The tension in the horizontal bar is:

T = 184.8 N

(b) The normal forces at A and B are:

NA = 223.2 N , NB = 453 N

(c) The components of the reaction force at the single hinge C are:

R2y = 453 N , R2x = 184.8 N

Note: Sorry for anyy mistake I could have committed.

I hope it helps.

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