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Chapter 10: Problem 1

A steel beam is placed vertically in the basement of a building to keep the floor above from sagging. The load on the beam is $5.8 \times 10^{4} \mathrm{N},\( the length of the beam is \)2.5 \mathrm{m},$ and the cross- sectional area of the beam is \(7.5 \times 10^{-3} \mathrm{m}^{2}\) Find the vertical compression of the beam.

Short Answer

Expert verified
Answer: The vertical compression of the steel beam is approximately \(0.000096625\) meters.

Step by step solution

01

Write down the given information

We know the following values: Load on the beam (F) = \(5.8 \times 10^{4} \mathrm{N}\) Length of the beam (L) = \(2.5 \mathrm{m}\) Cross-sectional area of the beam (A) = \(7.5 \times 10^{-3} \mathrm{m}^{2}\)
02

Find the stress

Using the formula for stress, we have: Stress (σ) = \(\frac{F}{A}\) Plug in the values: σ = \(\frac{5.8 \times 10^{4} \mathrm{N}}{7.5 \times 10^{-3} \mathrm{m^2}}\) σ = \(7.73 \times 10^6 \mathrm{Pa}\)
03

Find the strain

Using Hooke's Law, we have: Strain (ε) = \(\frac{σ}{E}\) The modulus of elasticity of steel (E) is approximately \(2 \times 10^{11} \mathrm{Pa}\). Plug in the values: ε = \(\frac{7.73 \times 10^6 \mathrm{Pa}}{2 \times 10^{11} \mathrm{Pa}}\) ε = \(3.865 \times 10^{-5}\)
04

Calculate the vertical compression

Using the formula for strain, we have: ε = \(\frac{ΔL}{L}\) We need to find ΔL (the vertical compression), so we rearrange the formula: ΔL = ε × L Plug in the values: ΔL = \(3.865 \times 10^{-5} \times 2.5 \mathrm{m}\) ΔL = \(9.6625 \times 10^{-5} \mathrm{m}\) Therefore, the vertical compression of the steel beam is approximately \(9.6625 \times 10^{-5}\) meters or \(0.000096625\) meters.

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Most popular questions from this chapter

A sewing machine needle moves with a rapid vibratory motion, rather like SHM, as it sews a seam. Suppose the needle moves \(8.4 \mathrm{mm}\) from its highest to its lowest position and it makes 24 stitches in 9.0 s. What is the maximum needle speed?
It takes a flea \(1.0 \times 10^{-3}\) s to reach a peak speed of $0.74 \mathrm{m} / \mathrm{s}$ (a) If the mass of the flea is \(0.45 \times 10^{-6} \mathrm{kg},\) what is the average power required? (b) Insect muscle has a maximum output of 60 W/kg. If \(20 \%\) of the flea's weight is muscle, can the muscle provide the power needed? (c) The flea has a resilin pad at the base of the hind leg that compresses when the flea bends its leg to jump. If we assume the pad is a cube with a side of \(6.0 \times 10^{-5} \mathrm{m},\) and the pad compresses fully, what is the energy stored in the compression of the pads of the two hind legs? The Young's modulus for resilin is $1.7 \times 10^{6} \mathrm{N} / \mathrm{m}^{2} .$ (d) Does this provide enough power for the jump?
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