Question 3.9: Four tugboats are used to bring an ocean liner to its pier. ......
Four tugboats are used to bring an ocean liner to its pier. Each tugboat exerts a 5000-lb force in the direction shown. Determine ( a ) the equivalent force couple system at the foremast O, (b) the point on the hull where a single, more powerful tugboat should push to produce the same effect as the original four tugboats.
a. Force-Couple System at O . Each of the given forces is resolved into components in the diagram shown (kip units are used). The force-couple system at O equivalent to the given system of forces consists of a force R and a couple {M}_{O}^{R} defined as follows:
R \ = \ \Sigma {F} \\ \\ ~~~~=\,(2.50\mathrm{i}\,-\,4.33\mathrm{j})\,+\,(3.00\mathrm{i}\,-\,4.00\mathrm{j})\,+\,(-5.00\mathrm{j})\,+\,(3.54\mathrm{i}\,+\,3.54\mathrm{j}) \\ \\ ~~~~=\;9.04\mathrm{i}\;-\;9.79\mathrm{j}{M}_{O}^{R}\,=\,{{{\Sigma}}}({r}\,\times\,{F}) \\ \\ ~~~~ =\,(-90\mathrm{i}\,+\,50\mathrm{j})\,\times\,(2.50\mathrm{i}\,-\,4.33\mathrm{j}) \\ \\~~~~ ~~+\ \ (100\mathrm{i}\ +\ 70\mathrm{j})\ \times\ (3.00\mathrm{i}\ -\ 4.00\mathrm{j})\\ \\ ~~~~ ~~ +~(400\mathrm{i}~+~70\mathrm{j})~\times~(-5.00\mathrm{j}) \\ \\ ~~~~~~ +\ (300\mathrm{i}\ +\ 70\mathrm{j})\times(3.54\mathrm{i}\ +\ 3.54\mathrm{j}) \\ \\ ~~~~=(390\,-\,125\,-\,400\,-\,210\,-\,2000\,+\,1062\,+\,248)\mathrm{k} \\ \\ ~~~~=\;-1035\mathrm{k}
The equivalent force-couple system at O is thus
{ R}\,=\,(9.04\;\mathrm{kips})\mathrm{i}\;-\;(9.79\;\mathrm{kips})\mathrm{j}\qquad{ M}_{0}^{R}\,=\,-(1035\;\mathrm{kip}\;\cdot\,\mathrm{ft})\mathrm{k}or {R}\,=\,13.33\,\,\mathrm{kips}\,~~\mathrm{c}\,\,47.3^{\circ}\,\,\,\,\,\,\,\,\,\,\,\,{ M}_{\small O}^{R}\,=\,1035\,\,\mathrm{kip}\,\cdot\,\mathrm{ft}\,\,\mathrm{i}
Remark. Since all the forces are contained in the plane of the figure, we could have expected the sum of their moments to be perpendicular to that plane. Note that the moment of each force component could have been obtained directly from the diagram by first forming the product of its magnitude and perpendicular distance to O and then assigning to this product a positive or a negative sign depending upon the sense of the moment.
b. Single Tugboat. The force exerted by a single tugboat must be equal to R , and its point of application A must be such that the moment of R about O is equal to { M}_{ O}^{R}. Observing that the position vector of A is
\mathrm{r}=x\mathrm{i}+70\mathrm{j}we write
{r}\times{R}={ M}_{ O}^{R} \\ \\ (x\mathrm{i}\ +\ 70\mathrm{j})\ \times\ (9.04\mathrm{i}\ -\ 9.79\mathrm{j})\,=\,-\,1035\mathrm{k} \\ \\ -x(9.79)\mathrm{k}\,-\,633\mathrm{k}\,=\,-1035\mathrm{k}\quad\quad\quad\quad\quad\quad\quad x\,=\,41.1~\mathrm{k}
