A stepladder of negligible weight is constructed as shown in Figure P12.57 where

Question

A stepladder of negligible weight is constructed as shown in Figure P12.57 where x-2.10 m. A painter of mass 66.0 kg stands on the ladder 3.00 m from the bottom. 2.00 m 3.00 m 2.00 m Figure P12.57 Assuming the floor is frictionless, find the following. (Suggestion: Treat the ladder as a single object, but also each half of the ladder separately.) (a) the tension in the horizontal bar connecting the two halves of the ladder (b) the normal forces at A and B at A) (at B) (c) 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

answers are:

a)139N

b)

a=404

b=242

c) 139

explanation:

as there is no friction at the floor, ,the reactions at A and B must are vertical.

and the ladder is of "negligible weight,"

part b)::

the vertical reactions sum to the painters weight:

Fa + Fb = 66.0kg * 9.8m/s² = 646 N

hence Summing the moments about B, we get

M = 0 = 66.0kg * 9.8m/s² * (5/8)*2.10m - Fa*2.10m

Fa = 404 N (b)

so Fb = 646N - Fa = 646-404=242 N (b)

Cut the ladder into vertically in half and consider the right side.

Sum the moments about the midpoint of the bar

-- any forces in the bar create no moment about that point.

M = 0 = Fb*1.1m - Fch*2.00m*sin

where Fch is the horizontal force at C (any vertical component has no moment action)

and = arccos(1.1/2.00) = 56.6º. So

0 = 242N * 1.1m - Fch*2.00m*sin56.6º

Fch = 139 N part of (c) -- thus the left ward component of right half on left = right ward component of left half on right (by Newton III)

Since there is no horizontal force at the floor, the tension in the rod must be

Frod = 139 N part (a)

Then analyzing the vertical forces (still on the right side of the ladder) gives

Fcy = -Fb = 242 N vertical part of (c), upward

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