- #1
Soumalya
- 183
- 2
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,
ΣFx=0 ΣFy=0 and ΣFz=0
The scalar moment equilibrium equations about three mutually perpendicular axes through a point,i.e,
ΣMx=0 ΣMy=0 and ΣMz=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 Txy and Tz respectively.The horizontal component Txy can then be resolved into components along x- and y- directions as Tx and Ty 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,
ΣFz=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,
ΣMx=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 ΣFz=0.
The moment equilibrium equation about the y-axis through O gives,
ΣMy=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]