Figure 12-57 shows an approximate plot of stress versus...

Figure $12-57$ shows an approximate plot of stress versus strain for a spider-web thread, out to the point of breaking at a strain of $2.00 .$ The vertical axis scale is set by values $a=0.12$ GN/m $^{2}, b=0.30$ GN/m $^{2}$ , and $c=0.80$ GN/m? Assume that the thread has an initial length of $0.80 \mathrm{cm},$ an initial cross-sectional area of $8.0 \times 10^{-12} \mathrm{m}^{2},$ and (during stretching) a constant volume. The strain on the thread is the ratio of the change in the thread's length to that initial length, and the stress on the thread is the ratio of the collision force to that initial cross-sectional area. Assume that the work done on the thread by the collision force is given by the area under the curve on the graph. Assume also that when the single thread snares a flying insect, the insect's kinetic energy is transferred to the stretching of the thread. (a) How much kinetic energy would put the thread on the verge of break- ing? What is the kinetic energy of (b) a fruit fly of mass 6.00 $\mathrm{mg}$ and speed 1.70 $\mathrm{m} / \mathrm{s}$ and $(\mathrm{c})$ a bumble bee of mass 0.388 $\mathrm{g}$ and speed 0.420 $\mathrm{m} / \mathrm{s} ?$ Would (d) the fruit fly and (e) the bumble bee break the thread?

Figure $12-57$ shows an approximate plot of stress versus strain for a spider-web thread, out to the point of breaking at a
strain of $2.00 .$ The vertical axis scale is set by values $a=0.12$
GN/m $^{2}, b=0.30$ GN/m $^{2}$ , and $c=0.80$ GN/m? Assume that the thread has an initial length of $0.80 \mathrm{cm},$ an initial cross-sectional area of
$8.0 \times 10^{-12} \mathrm{m}^{2},$ and (during stretching) a constant volume. The
strain on the thread is the ratio of the change in the thread's
length to that initial length, and the stress on the thread is the ratio of the collision force to that initial cross-sectional area.
Assume that the work done on the thread by the collision force is
given by the area under the curve on the graph. Assume also that
when the single thread snares a flying insect, the insect's kinetic energy is transferred to the stretching of the thread. (a) How
much kinetic energy would put the thread on the verge of break-
ing? What is the kinetic energy of (b) a fruit fly of mass 6.00 $\mathrm{mg}$
and speed 1.70 $\mathrm{m} / \mathrm{s}$ and $(\mathrm{c})$ a bumble bee of mass 0.388 $\mathrm{g}$ and speed 0.420 $\mathrm{m} / \mathrm{s} ?$ Would (d) the fruit fly and (e) the bumble bee break the thread?

Key Concepts

Stress?Strain Curve

The stress-strain curve graphically represents the relationship between the applied stress and the resulting strain in a material. This curve is fundamental in material science and engineering as it provides insight into the elastic and plastic behavior, yield point, ultimate strength, and the energy absorbed prior to failure, indicated by the area under the curve.

Constant Volume Deformation

Assuming constant volume during deformation implies that as a material is stretched and its length increases, its cross-sectional area decreases proportionally. This concept is important for accurately determining changes in geometry which affect stress calculations and the distribution of strain throughout the material.

Energy Conservation

The principle of energy conservation is applied in problems where kinetic energy, such as that from an impacting insect, is converted into mechanical work in deforming a material. By equating the kinetic energy with the work done in stretching the material, one can determine whether the energy provided is sufficient to cause failure.

Strain

Strain is a measure of deformation representing the relative change in length of a material when subjected to stress. It is calculated as the change in length divided by the original length. Recognizing strain helps in understanding how much a material stretches before reaching its elastic limit or breaking.

Stress

Stress is defined as the force applied per unit area within a material. It quantifies the intensity of internal forces that develop in response to an externally applied load. Understanding stress is crucial in analyzing how materials like threads or fibers respond to loads and when they may reach a failure point.

Work and Energy in Deformation

The work done on a material during deformation is equivalent to the energy stored in it, often referred to as elastic potential energy. In a stress-strain graph, this work is represented by the area under the curve. This concept links mechanical work with material deformation, highlighting how energy from external sources is absorbed and stored or dissipated by the material.

Transcript

00:01 So we know here, let's, we know that the force is equaling to a stress times an area.

00:09 We also know that the displacement is equaling to a strain times the total length.

00:24 So we can say that the work integral would be equalling to the integral of force times dx.

00:32 This is going to be equal to the integral of stress times the area times the differential strain times l and so we can say that the work would be equaling the area times the original length multiplied by the integral of the stress times the differential strain and we can then say that this means that the work is simply going to be equal to the the cross -sectional area of the thread, multiplied by the length of the thread, multiplied by the graph area under the curve.

01:26 So the stress, the integral of the stress times a differential strain, this essentially is the area under curve.

01:40 So here, the area under the curve, we can say that, we can simply say area.

01:57 Under curve this would be equal to one -half a times s sub one plus one -half times a plus b multiplied by s -2 minus s -sub -1 plus one -half times b plus one -half times b plus c multiplied by s -sub -3 minus s -sub -2 and then again area under curve this would be equal to one -half so factor out that one -half times a times s sub two plus b multiplied by s -3 minus s -sub -1 plus c times s -sub -3 minus s -sub -2 and we can solve so the area under curve would be equal to one -half and we know that the we know the units.

03:06 Stress is always going to be nons per square meter, and strain is, of course, unitless.

03:12 So we'll just lose the units completely.

03:15 Just a save space.

03:18 So 0 .12 times 10 to the 9th, multiplied by 1 .4, and then plus 0 .30 times 10 to the 9th, multiplied by 1 .0, plus 0 .80 times 10 to the 9th times 0 .60.

03:47 And we find that the area under the curve is going to be equal to 4 .74 times 10 to the 8th newtons per square meter.

04:04 So this would be, this is not really any answer.

04:08 However, we need to use this in order to calculate all of our subsequent parts.

04:14 So for part a, we can finally say that the kinetic energy that would put the thread on the verge of breaking is simply equal to the work.

04:23 So this would be equal to al times the, we can say graph area or again the area under the curve that we had just calculated.

04:33 And so the area is going to be equal to 8 .0 times 10 to the negative 12 square meters multiplied.

04:46 By 8 .0 times 10 to the negative third meters and then multiplied by the graph area 4 .74 times 10 to the 8th newtons per meter squared.

05:03 So the kinetic energy would be equal to 3 .03 times 10 to the 5th joules.

05:11 This would be our answer for part a.

05:15 For part b, the kinetic energy of the fruit fly of mass 6 .00 milligrams and a speed of 1 .70 meters per second, we can say that the kinetic energy would be equal to 1 .5 times mv squared...

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