Question 3.4: A rectangular plate is supported by brackets at A and B and ......
A rectangular plate is supported by brackets at A and B and by a wire CD . Knowing that the tension in the wire is 200 N, determine the moment about A of the force exerted by the wire on point C .
The moment {M}_{\mathrm{{A}}} about A of the force F exerted by the wire on point C is obtained by forming the vector product
{M}_{\mathrm{{A}}}={r}_{C/\mathrm{{A}}}\times\mathrm{{~F}} (1)
where {r}_{C/\mathrm{{A}}} is the vector drawn from A to C ,
{ r}_{C/A}={{\overrightarrow{{A C}} }}\:=\:(0.3\:{\mathrm m}){\mathrm i}\:+\:(0.08\:{\mathrm m}){\mathrm k} (2)
and F is the 200-N force directed along CD . Introducing the unit vector \mathrm{L}={\overrightarrow{{C D}} }/C D, we write
{F}=F{L}=(200\,{N})\,\frac{{{\overrightarrow{{C D}} }}}{C D} (3)
Resolving the vector {{{\overrightarrow{{C D}} }}} into rectangular components, we have
{{\overrightarrow{{C D}} }}\,=\,-(0.3\:\mathrm{m})\mathrm i\,+\,(0.24\:\mathrm{m})\mathrm j\,-\,(0.32\:\mathrm{m})\mathrm k\qquad C D\,=\,0.50\:\mathrm{m}
Substituting into (3), we obtain
{ F}=\frac{200\,{\mathrm N}}{0.50\,{\mathrm m}}\,[-(0.3\,{\mathrm m}){\mathrm i}\,+(0.24\,{\mathrm m}){\mathrm j}\,-\,(0.32\,{\mathrm m}){\mathrm k}] \\ \\ ~~~~~~~~~~ =\;-\;(\;120\mathrm{~N~})\mathrm i\,+\,(96\mathrm{~N~}){{\mathrm{j}}}\,-\,(\;128\mathrm{~N~}){\mathrm{k}} (4)
Substituting for {r}_{C/A} and F from (2) and (4) into (1) and recalling the relations (3.7) of Sec. 3.5, we obtain
{M}_{A}\,=\,{r}_{C/A}\,\times{F}\,=\,(0.3{\mathrm i}\,+\,0.08\mathrm{k})\,\times\,(-120 {\mathrm i}\,+\,\ 96 {\mathrm j}\,-\,128 {\mathrm{k}})\\ \\ ~~~~~~~~~~~~=\,(0.3)(96){\mathrm k}+\,(0.3)(-128)(-\mathrm j)\,+ (0.08)(-120)\mathrm{j}\,+\,(0.08)(96)(-\mathrm{i})\\ \qquad \qquad {M}_{A}\,=\,-(7.68\ \mathrm{N}\cdot\ \mathrm m)\mathrm{i}_{}^{\mathrm{}}\,+\,(28.8\ \mathrm{N}\cdot\ \mathrm m)\mathrm{j}\,+\,(28.8\ \mathrm{N}\cdot\ \mathrm m)\mathrm{k}Alternative Solution. As indicated in Sec. 3.8, the moment {M}_{A} can be expressed in the form of a determinant:
\qquad {M}_{A}=\begin{vmatrix} \mathrm i & \mathrm j & \mathrm k \\ x_{C}\,-\,x_{A} & y_{C}\,-\,y_{A} & z_{C}\,-\,z_{A} \\ \\ F_{x} & F_{y} & F_{z}\end{vmatrix}=\begin{vmatrix} \mathrm i & \mathrm j & \mathrm k \\ 0.3 & 0 & 0.08 \\ -120 & 96 & -128 \end{vmatrix}\\ \qquad \qquad {M}_{A}=-(7.68\, \mathrm {N}\cdot\mathrm {m}){\mathrm {i}}\,+\,(28.8\,\mathrm {N}\cdot\mathrm {m}){\mathrm {j}}\,+\,(28.8\,\mathrm {N}\cdot\mathrm {m})\mathrm {k}
