Answer:
FN₁ = 1146.6N : Force exerted on the ladder by the floor , vertical and upward
FN₂ = 407.5 N : Force exerted on the ladder by the wall , horizontal and opposite to friction force between the floor and the ladder
Explanation:
The equilibrium equation are:
∑Fx=0
∑Fy=0
∑M = 0
M = F*d
Where:
∑M : Algebraic sum of moments
M : moment ( N*m)
F : Force ( N)
d :Perpendicular distance of the force to the point ( meters )
Data
m =45 kg : mass of the ladder
M =72 kg : mass of the fire fighter
g = 9.8 m/s²: acceleration due to gravity
L = 12 m : ladder length
h = 9.3 m: ladder height
L/3 = 12/3 = 4m Location of the center of mass of the ladder of the way up
L/2 = 12/2 = 6m Location of the center of mass of the fire fighter
µ = 0 : coefficient of friction between the ladder and the wall
θ : angle that makes the ladder with the floor
sinθ = h/L = 9.3 m/12 m
θ =sin⁻¹( 9.3 / 12)
θ = 50.8°
Forces acting on the ladder
W₁ =m*g = 45 kg* 9.8 m/s² = 441 N: Weight of the ladder (vertical downward)
W₂ =M*g = 72 kg * 9.8 m/s² = 705.6 N : Weight of the fire fighter(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)
∑Fy=0
FN₁ -W₁ -W₂= 0
FN₁ = W₁ + W₂
FN₁ = 441N+ 705.6N
FN₁ = 1146.6N : Force exerted on the ladder by the wall (vertical and upward)
Calculation of the distances of the forces at the point A (contact point of the ladder on the floor)
d₁ = 4*cos 50.8° (m) = 2.53 m: Distance from W₁ to the point A
d₂ =6*cos 50.8° (m)= 3.79 m : Distance from W₂ to the point A
d₃ = 9.3 m : Distance from FN₂ to the point A
The 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(9.3) = (441 )(2.53) + (705.6)( 3.79 )
FN₂(9.3) = 1115.73 + 2674.2
FN₂ = (3790) / (9.3)
FN₂ = 407.5 N : Force exerted on the ladder by the wall , horizontal and opposite to friction force between the floor and the ladder
