Chapter 8.2, Problem 12P

To determine

To calculate: Ratio of loads the two beams can support

Answer to Problem 12P

Ratio of loads of the two beams is $\text{4:9}$ .

Explanation of Solution

Given information:

The following variation:

The load a beam with a rectangular cross section can support is jointly proportional to the beam’s width and square of its depth and inversely proportional to is length.

One beam is twice as wide, twice as long and three times as deep as a second beam.

Calculation:

For the first beam,

Let $L_1$be load, w be width, d be depth and l be length of the first beam. Then, L varies jointly as w and $d^2$ and inversely as l ,

  $L_1=k\frac{w{d}^2}{l}$

where k is a constant.

For the second beam,

Width is $w/2$ , length is $2l$ and depth is $3d$ . Therefore, load is,

  $\begin{array}{l}{L}_2=k\frac{\left(w/2\right){\left(3d\right)}^2}{\left(2l\right)}[\text{Equation]}\\ L_2=\frac{k\frac{w}{2}9d^2}{2l}[\text{Find square]}\\ L_2=\frac{9kw{d}^2}{4l}[\text{Simplify]}\end{array}$

Ratio of the loads is:

  $\begin{array}{l}\frac{L_1}{L_2}=\frac{\frac{kw{d}^2}{l}}{\frac{9kw{d}^2}{4l}}[\text{Ratio]}\\ \frac{L_1}{L_2}=\frac{\cancel{k}\cancel{w}\cancel{d^2}}{\cancel{l}}\times \frac{4\cancel{l}}{9\cancel{k}\cancel{w}\cancel{d^2}}[\text{Multiply by reciprocal]}\\ \frac{L_1}{L_2}=\frac{4}{9}[\text{Simplify]}\end{array}$