The uniform pole has a weight of 30 lb and a length of 26 ft. The coefficient of static friction between the floor and the pole is μs = 0.24.Determine the maximum distance d it can be placed from the smooth wall and not slip.

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General guidance

Concepts and reason

Force is a vector quantity. It is represented by a magnitude, direction, and point of application.

Frictional force which opposes the motion of a body. Here, the type of friction is dry friction which opposes the relative lateral motion between two solid surfaces in contact. Dry friction is of two types; static friction between stationary surfaces and kinetic friction between moving surfaces.

The major assumption for applying these equilibrium equations is that the body remains rigid.

Before applying these equilibrium equations, distinction should be made between the known and unknown forces that act on the body. When all the supports are removed by replacing them with forces that prevents the translation of body in a given direction, that diagram is called free body diagram.

Draw the free body diagram of the pole. Calculate the unknown forces using equilibrium equations. Use moment equilibrium to find the unknown angle. Find the distance.Fundamentals

The magnitude of frictional forces can be calculated as,

f=μN

Here, the coefficient of friction (either static or dynamic) is μ and the normal reaction is N.

A body is in equilibrium if vector sum of all the forces is equal to zero or moment of all forces about any point is equal to zero. That is,

Summation of all forces in x direction is zero, ∑Fx​=0

Summation of all forces in y direction is zero, ∑Fy​=0

Moment about arbitrary point is zero, ∑MO​=0

Draw the free body diagram of the pole.

Represent the normal and friction forces at the walls.

Equate the forces along vertical direction.

Calculate the maximum frictional force acting at the point A.

(FA​)max​=μs​NA​=(0.24)(30lb)=7.2lb

Equate the forces along horizontal direction.

∑​Fx​=0NB​−FA​=0NB​=7.2lb

For the pole to be balanced the forces in horizontal and vertical direction should equal. By using the force balance in x and y-direction the reaction forces are calculated.

Take moment about the point A.

∑​MA​=0(30lb)(13ft)cosθ−NB​(26ft)sinθ=0(30lb)(13ft)cosθ−(7.2)(26ft)sinθ=0θ=64.35∘​

Calculate the maximum distance pole can be placed from the smooth wall.

d=(26ft)cosθ=(26ft)cos64.35∘=11.25ft

Therefore, the maximum distance the pole can be placed from the smooth wall without slipping 11.25ft .

Using the moment balance angle made by the pole is calculated. Resolving the distance in horizontal direction, the maximum distance is calculated.