A thin stiff uniform rectangular plate with width L (L =AB) is lying on two inclined surface as shown in Fig. 3-1. The angle between the horizontal surface and the left inclined surface is α, and that between the horizontal surface and the right inclined surface is β. It is assumed that the plate length is sufficiently larger than L and both edges of the plate are smooth enough to neglect friction.
(1) In case of α+β = 90°, the static balance shown in Fig.3-1 is
(a) possible only provided α≠β
(b) always possible
(c) always impossible
The Attempt at a Solution
From the picture, at point A and B,
Calculating in horizontal axis : NA(normal force at a)*sinα = NBsinβ
vertically direction : NAcosα+NBcosβ=mg
And I took a moment at point B, so NA*L = mg*Lcosα.
And I tried to substitue all variables into α and β only, but I got quadratic equation that looked like wrong : (cosα)^2+cosα(sinα)/sinβ-2=0 because I have solved with x=-b+-b^2.... but it couldn't be solved.
Which way should I do to solve this problem?