All 8 fl long ladder has seven rungs. the rungs are spaced I ft apart, and the t

Question

All 8 fl long ladder has seven rungs. the rungs are spaced I ft apart, and the top and bottom rungs are 1 ft from their respective ends of the ladder. the top of the ladder has a roller. Neglect the weight of the ladder and assume the worker's hand applies no force to the ladder. the person weighs 200 lbs. For the next 3 questions, consider the person is standing on the middle rung of the ladder (as shown). Determine the MAGNITUDE of the friction force at location A. What direction does the friction force point at location A? Determine the coefficient of friction at location A. For the next 3 questions, consider the person is standing on the 6 h rung from the bottom (6 ft up the ladder). Determine the magnitude of the friction force at location A. Determine the coefficient of friction at location A. Given the coefficients of friction calculated in questions 3 and 5, is the ladder more likely to slip if someone is standing near the top of the ladder or near the bottom of the ladder? For the remaining questions, consider the system shown below. the pulley has a diameter of 0.25 meters and the length of the member is 1 meter from point A to point C. Location B is at the midpoint between A and C. It is of interest to determine the magnitude of the normal force acting on the member below point B. To perform this analysis, 4 different free body diagrams have been drawn (on back side of page). One or more of these free body diagrams is drawn incorrectly, and it is your goal to determine which of these diagrams are incorrect. Determine the magnitude of the normal force, N, when analyzing FBD I. Determine the magnitude of the normal force, N, when analyzing FBD II. Determine the magnitude of the normal force, N, when analyzing FBD III. Determine the magnitude of the normal force, N, when analyzing FBD IV. Based on your answers, which of the FBD's are drawn incorrectly? Why?

Explanation / Answer

forces acting on the ladder:

a) weight of the person , vertically downward

magnitude=200 lbs

force acting at 4 ft along the ladder from the bottom

b)normal force at A, from ground , in vertically upward direction

let magnitude be F1.

c)friction force at A,

acting to the right side, horizontally

let magnitude be F2

d)normal force at B

acting to the left side, horizontally

let magnitude be F3.

balancing forces in horizontal direction:

F2=F3...(1)

balancing forces in vertical direction:

F1=200 lbs...(2)

balancing torque about A:

[note: torque=force*perpendicular distance between the point where force is being applied

and the point about which torque is to be calculated.

as F1 and F2 are acting at A, their perpendicular distance=0

torque due to F1=torque due to F2 =0

as 200 lbs is acting at 4 ft along the ladder , its distance from point A

perpendicular direction=4*cos(60)=2 ft

as F3 is acting at point B which is 8 ft along the ladder from A,

its perpendicular distance =8*sin(60)=6.9282 ft]

hence torque balance equation becomes:

200*4*cos(60)=F3*8*sin(60)

==>F3=57.735 lbs

hence friction force at A is 57.735 lbs.

part 2:

friction force is acting to the right side as the ladder will slide to left side if it slides

hence friction force opposes the attempt of sliding and hence acts in the opposite direction to the direction of sliding

i.e. to the right side.

part 3:

coefficient of friction=friction force/normal force

=57.735/200=0.2886

part 4:

now, the only thing that has changed is distance of the person from point A becomes 6 ft

hence perpendicular distance of the point at which weight of the person is acting from point A =6*cos(60)=3ft

writing torque balance equation:

200*6*cos(60)=F3*8*sin(60)

==>F3=86.6025 lbs

part 5:

coefficient of friction at A=friction force/normal force

=86.6025/200=0.433

part 6:

as greater coefficient of friction is required to prevent sliding when the person is standing near to the top of the ladder,

it is more likely to slip when a person is standing near to the top of the tower.
then

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