Statics: Problem about Equilibrium in 3-dimensions

[Soumalya]

Homework Statement

The two uniform rectangular plates each weighing 800 kg are freely hinged about their common edge and suspended by the central cable and four symmetrical corner cables. Calculate the tension T in each of the corner cables and the tension T₀ in the center cable.

All dimensions in the figure are in meters.2. Homework Equations

The scalar force equilibrium equations in three mutually perpendicular directions x-,y- and z-,i.e,

ΣF_x=0 ΣF_y=0 and ΣF_z=0

The scalar moment equilibrium equations about three mutually perpendicular axes through a point,i.e,

ΣM_x=0 ΣM_y=0 and ΣM_z=0[/B]

The Attempt at a Solution

The Free Body Diagram of the plate assembly is drawn below along with the choice of the coordinate axes.

The tensions in the corner cables 'T' can be resolved into its horizontal and vertical components as T_xy and T_z respectively.The horizontal component T_xy can then be resolved into components along x- and y- directions as T_x and T_y respectively in the x-y plane.The angles which orient the line of action of a corner cable tension T can be determined as illustrated in the figure below.

The force equilibrium equations in the x- and y- directions are already satisfied since the identical x- and y- components of all the corner cable tensions cancel each other.

The force equilibrium equation in the z-direction yields,

ΣF_z=0

⇒4Tsinα+T₀-2.800(9.81)=0

⇒4T.$\frac {5}{\sqrt {46}}$+T₀=15696

⇒2.95T+T₀=15696

The moment equilibrium equation about the x-axis through O gives,

ΣM_x=0

⇒2Tsinα(6)+T₀(3)-2.800(9.81)(3)=0

⇒12T.$\frac {5}{\sqrt {46}}$+3T₀=47088

⇒8.85T+3T₀=47088

This is the same equation as obtained from ΣF_z=0.

The moment equilibrium equation about the y-axis through O gives,

ΣM_y=0

⇒ -2Tsinα(2$\sqrt {12}$)+800(9.81)($\sqrt {12}$+$\frac {\sqrt {12}} {2}$)-T₀($\sqrt {12}$)+800(9.81)($\frac {\sqrt {12}} {2}$)=0

⇒ -2T.$\frac {5} {\sqrt {46}}$.(2$\sqrt {12}$)+800(9.81)($\sqrt {12}$+$\frac {\sqrt {12}} {2}$+$\frac {\sqrt {12}} {2}$) -T₀($\sqrt {12}$)=0

⇒ 10.22T+$\sqrt {12}$T₀=54372.5
which is again the same equation as obtained before.

The moment equilibrium equation about the z-axis through O is already satisfied.
So as it is evident the equations of equilibrium are resulting into a single equation in T and T₀.We need at least two dissimilar equations in T and T₀ to solve for T and T₀.

Where is the problem?[/B]

 

[haruspex]

Where is the problem?

All of your equations would still be true if the pair plates were to form a single rigid body. You need one which depends on their being hinged.

 

[Soumalya]

All of your equations would still be true if the pair plates were to form a single rigid body. You need one which depends on their being hinged.

I had a thought about this approach earlier when a couple of things pushed me into doubts.Firstly,if we treat the two plates hinged together separately as two different bodies should we consider the effect of the centre cable on both the plates?

Secondly,if instead of being hinged together about their common edge the plates were welded together to form the assembly as shown(so that they could be treated as a single rigid body) would it be possible to determine the values of the tensions in that case using the conditions for static equilibrium?