Why does the traditional approach to this statics question not work?

Question

ABCD is a cross-section through the middle of a cube with side length a. The cube is attached to a point E above it on a smooth vertical wall EA by a light inextensible string also of length a fastened to the cube at B. Find the angle x between the string and the wall.

Traditional approach:

Let W be the weight of the cube, T be the tension in the string, R be the reaction at the wall.
Resolve vertically and horizontally:
T cos x = W (1)
T sin x = R
If you have drawn a decent diagram with T, R and W coincident at P and AP = p, then taking moments about A gives
Wp = Tpcos x
You get nowhere, as this is identical to (1) above.

Ironically if you draw a bad diagram with the forces not coincident, it forces you to delve further into the geometry and you get the answer, which is tan x = 1/3.

Sorry about the ambiguity. It wasn't intended to be a 3-d question. ABCD is parallel to a face of the cube. B is the mid-point of an edge.

E is vertically above B.

Answer 1

It's a 3d geometry question.


If ABCD is 'any' cross section (through the middle), then point B will lie on one of the edges of the cube.

Symmetry allows us to assume B is between one corner and the middle of the edge.

If B is on the middle of an edge, the cube will hang 'normally (ie. with edges horizontal) and thus a bottom edge will lie against the wall.

However if B is at a corner, the cube will hang 'point up' (with edges at 45% to the horizontal) and thus a lower corner point will lie against the wall and the cube will hang 'symmetrically', however I don't thing this position (point against the wall) is stable - it will 'flip' onto an edge position at the slightest disturbance.

At intermediate positions of B the cube will hang non-symmetrically

This suggest to me that the angle between string and wall must be a function of the exact position of B.

Answer 2

I'm sorry but I think this question requires a diagram which I know, of course, you can't supply. But I admire you if you can even understand the question without a diagram...if this the case.

Sorry.