Chapter 3.2, Problem 3.64P

To determine

Find the perpendicular distance (d) between rod AB and the line of action of P.

Answer to Problem 3.64P

The perpendicular distance (d) between rod AB and the line of action of P is $\underset{\_}{13.06\text{in}\text{.}}$.

Explanation of Solution

Given information:

The length of the vertical rod CD $\left(l_{CD}\right)$ is 23 in.

The length of the rod AB $\left(l_{AB}\right)$ is 50 in.

The force (P) is 235 lb.

The distance between the points O to B (x) is 32 in.

The distance between the points O to A (y) is 24 in.

The vertical height of OA $\left(h_{OA}\right)$ is 30 in.

The vertical height of OH $\left(h_{OH}\right)$ is 17 in.

The horizontal distance between point O and midpoint C $\left(d_{OC}\right)$ is 16 in.

The horizontal distance between midpoint C and point G over x axis $\left(d_{CG}\right)$ is 21 in.

The horizontal distance between point B and midpoint C $\left(d_{BC}\right)$ is 12 in.

The horizontal distance between midpoint C and point G over z axis (h) is 18 in.

Calculation:

Calculate the position vector of AB using the relation:

$\stackrel{\rightharpoonup }{AB}=xi-h_{OA}j-yk$

Substitute 32 in. for x, 30 in. for $h_{OA}$ and 24 in. for y.

$\begin{array}{c}\stackrel{\rightharpoonup }{AB}=32i-30j-24k\\ AB=\sqrt{{\left(32\right)}^2+{\left(-30\right)}^2+{\left(-24\right)}^2}\\ =50\text{in}\text{.}\end{array}$

Calculate the unit vector of AB $\left({\lambda }_{AB}\right)$ using the formula:

${\lambda }_{AB}=\frac{\stackrel{\rightharpoonup }{AB}}{AB}$

Substitute $32i-30j-24k$ for $\stackrel{\rightharpoonup }{AB}$ and 50 in. for AB.

$\begin{array}{c}{\lambda }_{AB}=\frac{32i-30j-24k}{50}\\ =0.64i-0.6j-0.48k\end{array}$

Apply the force P at the point G.

Calculate the position vector of from point G to point B $\left(r_{G/B}\right)$ using the relation:

$r_{G/B}=\left[x-\left(d_{OC}+d_{CG}\right)\right]i+\left(d_{BC}+h\right)k$

Substitute 32 in. for x, 16 in. for $d_{OC}$, 21 in. for $d_{CG}$, 12 in. for $d_{BC}$ and 18 in. for h.

$\begin{array}{c}{r}_{G/B}=\left[32-\left(16+21\right)\right]i+\left(12+18\right)k\\ =5i+30k\end{array}$

Calculate the position vector of DG using the relation:

$\stackrel{\rightharpoonup }{DG}=d_{CG}i-\left(\frac{h_{OA}}{2}+l_{CD}\right)j+hk$

Substitute 21 in. for $d_{CG}$, 30 in. for $h_{OA}$, 23 in. for $l_{CD}$ and 18 in. for h.

$\begin{array}{c}\stackrel{\rightharpoonup }{DG}=21i-\left(\frac{30}{2}+23\right)j+18k\\ =21i-38j+18k\\ DG=\sqrt{{\left(21\right)}^2+{\left(-38\right)}^2+{\left(18\right)}^2}\\ =47\text{in}\text{.}\end{array}$

Calculate the force $\left(P\right)$ using the formula:

$P=P\frac{\stackrel{\rightharpoonup }{DG}}{DG}$

Substitute 235 lb for P, $21i-38j+18k$ for $\stackrel{\rightharpoonup }{DG}$ and 47 in. for DG.

$\begin{array}{c}P=235\times \frac{21i-38j+18k}{47}\\ =235\times \left(0.447i-0.808j+0.383k\right)\\ =105i-189.9j+90k\\ \approx 105i-190j+90k\end{array}$

Calculate the position vector at P $\left({\lambda }_P\right)$ using the relation:

${\lambda }_P=\frac{P}{P}$

Substitute $105i-190j+90k$ for P and 235 lb for P.

$\begin{array}{c}{\lambda }_P=\frac{105i-190j+90k}{235}\\ =0.447i-0.808j+0.383k\end{array}$

Calculate the angle between AB and P $\left(\theta \right)$ using the formula:

$\cos \theta ={\lambda }_{AB}\cdot {\lambda }_P$

Substitute $0.64i-0.6j-0.48k$ for ${\lambda }_{AB}$ and $0.447i-0.808j+0.383k$ for ${\lambda }_P$.

$\begin{array}{c}\cos \theta =\left(0.64i-0.6j-0.48k\right)\cdot \left(0.447i-0.808j+0.383k\right)\\ =0.28608+0.4848-0.18384\\ =0.5870\end{array}$

$\begin{array}{c}\theta ={\cos }^{-1}\left(0.5870\right)\\ =54.04°\end{array}$

Calculate the moment about AB $\left(M_{AB}\right)$ using the relation:

$M_{AB}={\lambda }_{AB}\cdot \left(r_{G/B}\times P\right)$

Substitute $0.64i-0.6j-0.48k$ for ${\lambda }_{AB}$, $5i+30k$ for $r_{G/B}$ and $105i-190j+90k$ for P.

$\begin{array}{c}{M}_{AB}=\left|\begin{array}{ccc}0.64& -0.6& -0.48\\ 5& 0& 30\\ 105& -190& 90\end{array}\right|\\ =\left[0.64\left(0-\left(-5700\right)\right)\right]-\left[\left(-0.6\right)\left(450-3150\right)\right]-\left[0.48\left(-950-0\right)\right]\\ =3648-1620+456\end{array}$

$\begin{array}{c}=2484\text{lb}\cdot \text{in}\text{.}\times \frac{1}{12}\frac{\text{ft}}{\text{in}\text{.}}\\ =207\text{lb}\cdot \text{ft}\end{array}$

Calculate the perpendicular distance (d) between rod AB and the line of action of P by taking moment about AB:

$M_{AB}=\left(P\sin \theta \right)d$

Substitute 235 lb for P and $54.04°$ for $\theta$ and $207\text{lb}\cdot \text{ft}$ for $M_{AB}$.

$\begin{array}{c}207=\left(235\sin 54.04\right)d\\ d=\frac{207}{235\sin 54.04}\\ =1.088\text{ft}\times \text{12}\frac{\text{in}\text{.}}{\text{ft}}\\ =13.06\text{in}\text{.}\end{array}$

Therefore, the perpendicular distance (d) between rod AB and the line of action of P is $\underset{\_}{13.06\text{in}\text{.}}$.