Here
L = ladder length
P = ladder weight
F_a = applied horizontal force
mu = static friction coefficient
H = normal force in the vertical wall
V = normal force in the floor
y_0 = height ladder contact point
x_0 = width ladder contact point
d = applied force distance over the ladder
sin(theta) = y_0/sqrt(y_0^2+x_0^2)
Now we will develop the modeling for two cases:
1) F_a pushing to the left
{(sum_X->H - F_a + mu V = 0),(sum_Y-> V - P - mu H = 0),(M_O-> x_0 V + F_a d sin(theta) - P (x_0/2) - y_0 H = 0):}
2) F_a pushing to the right
{(sum_X->H + F_a - mu V = 0),(sum_Y-> V - P + mu H = 0),(M_O-> x_0 V - F_a d sin(theta) - P (x_0/2) - y_0 H = 0):}
Solving both cases we have for each case:
1) F_a = 1/2(P ((mu^2-1) x_0 - 2 mu y_0) sqrt[x_0^2 + y_0^2])/(
(mu x_0 - y_0) sqrt[x_0^2 + y_0^2]+d (1 + mu^2) y_0)
putting figures we obtain the condition F_a > 416.667 to the left
2) F_a = 1/2(P ((mu^2-1) x_0 + 2 mu y_0) sqrt[
x_0^2 + y_0^2])/(
(mu x_0 + y_0) sqrt[x_0^2 + y_0^2]- d (1 + mu^2) y_0)
putting figures we obtain the condition F_a > 38.889 to the right.
