Answer:
fs =202.3N
Explanation:
The equilibrium equation are:
∑Fx=0
∑Fy=0
∑M = 0
Where:
∑Fx :algebraic sum of forces in the direction of the x-axis
∑Fy: algebraic sum of forces in the direction of the y-axis
∑M : Algebraic sum of moments
M = F*d
M : moment ( N*m)
F : Force ( N)
d :Perpendicular distance of the force to the point ( meters )
Data
W₁ =138 N :weight of the ladder
W₂ =765 N : weight of the house painter
L = 5 m : ladder length
h = 4.7 m : ladder height
µ = 0 : coefficient of friction between the ladder and the wall
θ : angle that makes the ladder with the floor
sinθ = h/L = 4.7 m / 5 m
θ =sin⁻¹(4.7 / 5)
θ = 70.05°
Forces acting on the ladder
W₁ = 138 N :Weight of the ladder (vertical downward)
W₂ =765 N : Weight of the fire painter(vertical downward)
FN₁ :Normal force that the floor exerts on the ladder (vertical upward) (point A)
fs : friction force that the floor exerts on the ladder (horizontal and opposite the movement )(point A)
FN₂ : Normal Force that the wall exerts on the ladder ( horizontal and opposite to friction force between the floor and the ladder)
Calculation of the distances of the forces at the point A (contact point of the ladder on the floor)
d₁ =2.5 *cos 70.05° = 0.853 m: Horizontal distance from W₁ to the point A
d₂ = 3 / tan 70.05° = 1.089 m : Horizontal distance from W₂ to the point A
d₃ = 4.7 m : Vertical distance from FN₂ to the point A
Equilibrium equation of the moments at the point A (contact point of the ladder with the floor)
∑MA = 0
FN₂(d₃) - W₁( d₁) - W₂(d₂) = 0
FN₂(d₃) = W₁(d₁) + W₂(d₂)
FN₂( 4.7) = ( 138 )( 0.853) + (765)( 1.089 )
FN₂( 4.7) = 117.714 + 833.085
FN₂ = (950.799) / ( 4.7)
FN₂ =202.3 N : Force exerted on the ladder by the wall , horizontal and opposite to friction force between the floor and the ladder
Equilibrium equation of forces in x-direction
∑Fx=0
FN₂ - fs = 0
202.3 N - fs = 0
202.3 N = fs
fs = 202.3 N : force of friction that the driveway exerts on the bottom of the ladder
