Equilibrium of a System

Question

A triangular plate, vertices at points A, B and C, has uniformly distributed mass M and is held horizontally with the help of three vertical forces acting through the three vertices. If AB = c, BC = a and CA = b, find magnitudes of three forces in terms of Mg, a, b and c. Here g is magnitude of uniform gravitational field intensity and is supposed to be same at all points of the plate.

Found this problem on internet, no idea how to solve this but looks really interesting. Any help would be much appreciated.

Answer 1

I explain the approach.

There are three unknowns FA , FB, and FC which are forces acting on vertices A, B and C respectively.

Obviously

FA+FB+FC = Mg

Let a' be the distance between a and center of gravity and a'' be the distance between vertex A and a .

Imagine the following situation. you hold up the triangle from vertices B and C. Now if you want to keep the plate horizontally, Force A must overcome the torque with base at a and arm of rotation equal to a' .

Let's write the equation of torque in this situation,

FA a'' = Mg a'

Similarly, we get

FB b'' = Mg b'

and

FC c'' = Mg c'

Or.

FC c''/c' = FB b'' / b' = FA a'' / a' = Mg

Now we have 4 equations for three unknowns. So, we should be able to find FA. FB, and FC.