determine the magnitude of the force?

The contact of the sphere with each plane is a point with no area of contact. With each contact point, unless given otherwise, the force will always be perpendicular to each plane surface. So each force with have a horizontal component and a vertical component.

The simplest way to solve this is to look at the sphere force on each plane, instead of looking at the horizontal and vertical force equations on the sphere.

Looking at the sphere force on the 70 degree plane, which is the component of the sphere weight perpendicular to the 70 degree plane:

Sphere Force On 70 Degree Plane = m g cos(90 - 70 degrees) = (20 kg)(9.8 m/s^2) cos(20 degrees)

= 184 N

Looking at the sphere force on the 30 degree plane, which is the component of the sphere weight perpendicular to the 30 degree plane:

Sphere Force On 30 Degree Plane = m g cos(90 - 60 degrees) = (20 kg)(9.8 m/s^2) cos(30 degrees)

= 170 N

The key here is to draw the force triangles with the sphere weight forces being the hypotenuses of each triangle.

through fact there are no forces to counter act the pulling stress, all 3 blocks are in a state of acceleration (defined as F=MA). With the given setup, the acceleration of all 3 blocks could be same. look on the blocks and strings as 3 seperate platforms. device #a million: Block C being speeded up by potential of String BC device #2: Block B & C being acclerated by potential of String AB device #3: Block A, B, & C acclerated by potential of String A provided that F=MA, and A is an analogous for all 3 platforms, then the stress if string AB could be two times that of String BC through fact the 2d device has two times the mass. String A has three times the stress of String BC through fact the third device has three times the mass.