3D Equilibrium: Hanging Plate

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Name	Status	Author	Last Modified
3D Equilibrium: Hanging Plate	Ready to use	William Haynes	31/12/2023 13:50
Demo: 3D Geogebra diagram	Ready to use	William Haynes	07/02/2024 16:14

Question statement

{applet}

You can rotate the diagram and move the labels with the mouse, zoom with the mouse wheel, and shift-drag to pan.

A $\var{qty(L,mm)} \times \var{qty(W,mm)}$ steel plate $\var{qty(t,mm)}$ thick is suspended from point $P$ located $\var{qty(H,mm)}$ above the center of the plate.

Knowing that $DE = \var{qty(DE,mm)}$ and that the density of the steel is $\rho = \var{qty(rho, "kg/m^3")}$ , determine the tensions in the three cables.

Give any introductory information the student needs.

			Name	Type	Generated Value

W	integer	450
L	integer	440
H	integer	625
frac	number	0.4
rho	integer	7840
t	integer	20
applet	ggbapplet	HTML node
params	dict	Dictionary with 4 entries
g	quantity	quantity(9.81, "m/s^2")
mm	string	mm

			Name	Type	Generated Value

DE	number	270
CD	number	180
weight	quantity	quantity(304.565184, "N")

			Name	Type	Generated Value

r_A	vector	vector(220,-225,-625)
r_B	vector	vector(220,225,-625)
r_D	vector	vector(-220,45,-625)
u_A	vector	Vector with 3 components
u_B	vector	Vector with 3 components
u_D	vector	Vector with 3 components

			Name	Type	Generated Value

MA	matrix	Matrix of size 3×3
P	matrix	matrix([0],[0],[304.565184])
MC	matrix	Matrix of size 3×4
soln	matrix	Matrix of size 3×4

			Name	Type	Generated Value

T_A	number	102.2973551284
T_B	number	68.198233881
T_D	number	161.8132691405

			Name	Type	Generated Value

F_A	vector	Vector with 3 components
F_B	vector	Vector with 3 components
F_D	vector	Vector with 3 components
F_P	vector	vector(0,0,304.565184)
sum	vector	Vector with 3 components

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(400..800#50)

JME function reference

Description

width of plate in mm

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

450

→ Used by:

CD
DE
params
r_A
r_B
r_D
weight

Statement
Advice

Parts

Answers Gap-fill

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$T_A$ = $T_A$ $T_B$ = $T_B$ $T_D$ = $T_D$

Advice

To find the tensions in the cables, consider equilibrium of particle $P$, and note that force $\mathbf{P}$ acts vertically up with a magnitude equal to the weight of the plate.

Find the weight of the plate, and the upward force $\mathbf{P}$.

$\begin{align} W &= m g = \rho\ (L\times W \times t)\ g \\
& = \var{qty(rho, "kg/m^3")} \times \left(\var{qty(L,"mm")} \times \var{qty(W,"mm")} \times \var{qty(t,"mm")} \right) \times \left(\frac{\var{qty(1, "m")}}{\var{qty(1000,"mm")}}\right)^3 \times \var{g}\\
& = \var{siground(weight,4)}\\ \mathbf{P} &= \langle 0,0, \var{siground(scalar(weight),4)} \rangle \text{ N}\\ \end{align}$

Find position vectors and corresponding unit vectors from point $P$ to points $A$, $B$, and $D$.

$\begin{align}
r_A &= \var{ brackets(r_A)} \text{ mm} & u_A &= \frac{r_A}{|r_A|} = \var{brackets(U_A)} \\
r_B &= \var{ brackets(r_B)} \text{ mm} & u_B &= \frac{r_B}{|r_B|} = \var{brackets(U_B)} \\
r_D &= \var{ brackets(r_D)} \text{ mm} & u_D &= \frac{r_D}{|r_D|} = \var{brackets(U_D)} \\
\end{align}$

Express the forces in the cables in terms of unknown tensions and the unit vectors.

$\begin{align}
\mathbf{T_A} & = T_A u_A = T_A \var{brackets(u_A)}\\
\mathbf{T_B} & = T_B u_B = T_B \var{brackets(u_B)}\\
\mathbf{T_D} & = T_D u_A = T_D \var{brackets(u_D)}\\
\end{align}$

Set up three scalar equilibrium equations corresponding to $\mathbf{T_A} + \mathbf{T_B} + \mathbf{T_D} + \mathbf{P} = 0$.

\begin{align}
\Sigma F_x&=0: &\simplify[!zeroTerm]{+{precround(U_A[0],3)} T_A + {precround(U_B[0],3)} T_B + { precround(U_D[0],3)} T_D + 0 = 0 }\\
\Sigma F_y&=0: &\simplify[!zeroTerm]{+{precround(U_A[1],3) } T_A + {precround(U_B[1],3) } T_B + { precround(U_D[1],3) } T_D + 0 =0} \\
\Sigma F_z&=0: &\simplify[!zeroTerm]{{precround(U_A[2],3) } T_A + {precround(U_B[2],3) } T_B + { precround(U_D[2],3) } T_D + {siground(scalar(weight),4)} =0}\\
\end{align}

Solve the equilibrium equations simultaneously to get:

$\begin{align}
T_A &= \var{siground(qty(T_A,"N"),4)}\\
T_B &= \var{siground(qty(T_B,"N"),4)}\\
T_D &= \var{siground(qty(T_D,"N"),4)}\\
\end{align}$

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This question is used in the following exams:

Chapter 3 Exercises by William Haynes in Engineering Statics.
Homework #2 by Mallory Barkdull in Mallory's workspace.

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3D Equilibrium: Hanging Plate

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 1 year, 2 months ago

William Haynes 1 year, 2 months ago

William Haynes 1 year, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`