Chapter 12, Problem 073 Your answer is partially correct. Try again. A uniform a

Question

Chapter 12, Problem 073 Your answer is partially correct. Try again. A uniform adder is 15 m lo ng and weighs 220 N. In the figure, the ladder leans against a vertical, frictionless wall at height h-6.5 m above the ground. A horizontal force ce d 2.8 m from its base (measured along the ladder). If force magnitude F-46 N, what are (a)x, (b)y components of the force ground on the ladder? If F 160 N, what are (c), (d)y-components of the force of the ground on the ladder? (e) Suppose the coeficlent of static friction between the ladder and the ground is 0.38; for what minimum value of the force magnitude F will the base of the ladder just barely start to move toward the wall? L.

Explanation / Answer

First we need the angle w.r.t horizontal: = arcsin(h/L) = arcsin(6.5/15) = 25.68°

(a) If we sum the moments about the base, we don't have to worry about the normal or friction forces there. There is

no vertical (friction) force at the wall.

M = 0 = Fw x 6.5 - 220 x 15/2 x cos25.68° - 46 x 2.8 x sin25.68°

(where Fw is the normal force at the wall)

Fw = 237.36 N

Then Fx = 237.36 - 46 = 191.36 N (to the right)

(b) Fy = 220 N (up, the weight of the ladder)

(c) M = 0 = Fw x 6.5 - 220 x 15/2 x cos25.68° - 160 x 2.8 x sin25.68°

Fw = 258.64 N

Then Fx = 258.64 - 160 = 98.64 N (to the right)

(d) and simply Fy = 220 N (up, the weight of the ladder)

(e) the friction at the floor depends only on the normal force, which we've just demonstrated is independent of the horizontal load and is 220 N.

So max friction Ff = µ x Fy = 0.38 x 220 = 83.6 N

Then the ladder starts to move when Fx exceeds 83.6 N. Now we work it backwards:

F = Fx + Fw = 83.6 N + Fw

M = 0 = Fw x 6.5 - 220 x 15/2 x cos25.68° - F x 2.8 x sin25.68°

Substitute F = Fw + 83.6 N and do some calculations:

0 = Fw x 6.5 - 1487.03 - (Fw + 83.6) x 1.21

0 = 6.5 Fw - 1487.03 - 1.21Fw - 101.44

5.29 Fw = 1588.47

Fw = 300.28 N

and finally F = 300.28 - 83.6 = 216.68 N

Please rate my answer if you find it helpful, good luck...

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