Solut ion

The change in volume, found from equation 10.20, is

(0.500 m3)(3.00 X 10s N/m2)14 X 10,0N/m2

= -1.1 X 10-* m3

The minus sign indicates that the volume has decreased.

The Language of Physics

ElasticityThat property of a body by which itexperiences a change in size or shapewhenever a deforming force acts on thebody. The elastic properties of matter are amanifestation of the molecular forces thathold solids together (p. 283).Lattice structure of a solidA regular, periodically repeated, three-dimensional array of the atoms ormolecules comprising the solid (p. 283).StressFor a body that can be either stretched orcompressed, the stress is the ratio of theapplied force acting on a body to the cross-sectional area of the body (p. 285).StrainFor a body that can be either stretched orcompressed, the ratio of the change inlength to the original length of the body iscalled the strain (p. 285).Hooke's lawIn an elastic body, the stress is directlyproportional to the strain (p. 285).

Young's modulus of elasticityThe proportionality constant in Hooke'slaw. It is equal to the ratio of the stress tothe strain (p. 285).Elastic limitThe point where the stress on a bodybecomes so great that the atoms of thebody are pulled permanently away fromtheir equilibrium position in the latticestructure. When the stress exceeds theelastic limit, the material will not return toits original size or shape when the stress isremoved. Hooke's law is no longer validabove the elastic limit (p. 286).ShearThat elastic property of a body that causesthe shape of the body to be changed when astress is applied. When the stress isremoved the body returns to its originalshape (p. 288).Shearing strainThe angle of shear, which is a measure ofhow much the body's shape has beendeformed (p. 289).

Shearing stressThe ratio of the tangential force acting onthe body to the area of the body over whichthe tangential force acts (p. 289).Shear modulusThe constant of proportionality in Hooke'slaw for shear. It is equal to the ratio of theshearing stress to the shearing strain(p. 290).Bulk modulusThe constant of proportionality in Hooke'slaw for volume elasticity. It is equal to theratio of the compressional stress to thestrain. The strain for this case is equal tothe change in volume per unit volume(p. 291).Elasticity of volumeWhen a uniform force is exerted on allsides of an object, each side of the objectbecomes compressed. Hence, the entirevolume of the body decreases. When theforce is removed the body returns to itsoriginal volume (p. 291).

Summary of Important Equations

Hooke's law in generalstress oc strainHooke's law for stretchingor compressionL=Y±kA L 0

(10.5)

(10.6)

Hooke's law for a springF = kxHooke's law for shearFtA S<t>

(10.9)

(10.15)

Hooke's law for volumeelasticityF A V— = - B —A V 0Hooke's law for volumeelasticityP = " S T T

(10.19)

(10.20)

292 Vibratory Motion, Wave Motion, and Fluids

Questions for Chapter 10

1. Why is concrete often reinforcedwith steel?

+2. An amorphous solid such as glassdoes not have the simple latticestructure shown in figure 10.1. Whateffect does this have on the elasticproperties of glass?

3. Discuss the assumption that thediameter of a wire does not changewhen under stress.

4. Compare the elastic constants of ahuman bone with the elasticconstants of other materials listed intable 10.1. From this standpointdiscuss the bone as a structuralelement.

5. Why are there no Young's moduli forliquids or gases?

6. Describe the elastic properties of acube of jello.

7. If you doubled the diameter of ahuman bone, what would happen tothe maximum compressive force thatthe bone could withstand withoutbreaking?

+8. In the profession of Orthodontics, adentist uses braces to realign teeth.Discuss this process from the point ofview of stress and strain.

+9. Discuss Hooke's law as it applies tothe bending of a beam that is fixed atone end and has a load placed at theother end.

(a)

+ 10. How do the elastic properties of amaterial affect the vibration of thatmaterial?

Problems for Chapter 10

70.2 Hooke's Law-and Strain

-Stress

1. An aluminum wire has a diameter of0.850 mm and is subjected to a forceof 1000 N. Find the stress acting onthe wire.

2. A copper wire experiences a stress of5.00 X 103 N/m2. If the diameter ofthe wire is 0.750 mm, find the forceacting on the wire.

3. A brass wire 0.750 cm long isstretched by 0.001 cm. Find thestrain of the wire.

4. A steel wire, 1.00 m long, has adiameter of 1.50 mm. If a mass of3.00 kg is hung from the wire, byhow much will it stretch?

5. A load of 50,000 lb is placed on analuminum column 4.00 in. indiameter. If the column wasoriginally 4.00 ft high find theamount that the column has shrunk.

6. A mass of 25,000 kg is placed on asteel column, 3.00 m high and 15.0cm in diameter. Find the decrease inlength of the column under thiscompression.

7. An aluminum wire, 1.50 m long, hasa diameter of 0.750 mm. If a force of60.0 N is suspended from the wire,find (a) the stress on the wire, (b) theelongation of the wire, and (c) thestrain of the wire.

8. A copper wire, 1.00 m long, has adiameter of 0.750 mm. When anunknown weight is suspended fromthe wire it stretches 0.200 mm. Whatwas the load placed on the wire?

9. A steel wire is 1.00 m long and has adiameter of 0.75 mm. Find themaximum value of a mass that canbe suspended from the wire beforeexceeding the elastic limit of thewire.

10. A steel wire is 1.00 m long and has a10.0-kg mass suspended from it.What is the minimum diameter ofthe wire such that the load will notexceed the elastic limit of the wire?

11. Find the maximum load that can beapplied to a brass wire, 0.750 mm indiameter, without exceeding theelastic limit of the wire.

12. Find the maximum change in lengthof a 1.00-m brass wire, of 0.800 mmdiameter, such that the elastic limitof the wire is not exceeded.

13. If the thigh bone is about 25.0 cm inlength and about 4.00 cm in diameterdetermine the maximum compressionof the bone before it will break. Theultimate compressive strength ofbone is 1.70 X 108 N/m2.

14. If the ultimate tensile strength ofglass is 7.00 X 107 N/m2, find themaximum weight that can be placedon a glass cylinder of 0.100 m2 area,25.0 cm long, if the glass is not tobreak.

15. A human bone is 2.00 cm indiameter. Find the maximumcompression force the bone canwithstand without fracture. Theultimate compressive strength ofbone is 1.70 X 108N/m2.

16. A copper rod, 0.400 cm in diameter,supports a load of 150 kg suspendedfrom one end. Will the rod return toits initial length when the load isremoved or has this load exceededthe elastic limit of the rod?

70.3 Hooke's Law for a Spring17. A coil spring stretches 4.00 cm when

a mass of 500 g is suspended from it.What is the force constant of thespring

18. A coil spring stretches by 2.00 cmwhen an unknown load is placed onthe spring. If the spring has a forceconstant of 3.5 N/m, find the valueof the unknown force.

19. A coil spring stretches by 2.50 cmwhen a mass of 750 g is suspendedfrom it. (a) Find the force constant ofthe spring, (b) How much will thespring stretch if 800 g is suspendedfrom it?

20. A horizontal spring stretches 20.0 cmwhen a force of 10.0 N is applied tothe spring. By how much will itstretch if a 30.0-N force is nowapplied to the spring? If the samespring is placed in the vertical and aweight of 10.0 N is hung from thespring, will the results change?

21. A coil spring stretches by 4.50 cmwhen a mass of 250 g is suspendedfrom it. What force is necessary tostretch the spring an additional 2.50cm?

Chapter 10 Elasticity 293

70.4 Elast ici ty of Shape—Shear22. A brass cube, 5.00 cm on a side, is

subjected to a tangential force. If theangle of shear is measured in radiansto be 0.010 rad, what is themagnitude of the tangential force?

23. A copper block, 7.50 cm on a side, issubjected to a tangential force of 3.5X 103 N. Find the angle of shear.

24. A copper cylinder, 7.50 cm high, and7.50 cm in diameter, is subjected to atangential force of 3.5 X 103 N.Find the angle of shear. Comparethis result with problem 23.

25. An annular copper cylinder, 7.50 cmhigh, inner radius of 2.00 cm andouter radius of 3.75 cm, is subjectedto a tangential force of 3.5 X 103 N.Find the angle of shear. Comparethis result with problems 23 and 24.

70.5 Elasticity of Volume26. A cube of lead 15.0 cm on a side is

subjected to a uniform pressure of5.00 X 105 N/m2. By how muchdoes the volume of the cube change?

27. A liter of glycerine contracts 0.21cm3 when subject to a pressure of 9.8X 105 N/m2. Calculate the bulkmodulus of glycerine.

28. A pressure of 1.013 X 107 N/m2 isapplied to a volume of 15.0 m3 ofwater. If the bulk modulus of wateris 0.020 X 1010 N/m2, by how muchwill the water be compressed?

29. Repeat problem 28, only this timeuse glycerine that has a bulkmodulus of 0.45 X 10l0N/m2.

30. Normal atmospheric pressure is1.013 X 105 N/m2. How manyatmospheres of pressure must beapplied to a volume of water tocompress it to 1.00% of its originalvolume? The bulk modulus of wateris 0.020 X 10'»N/m2.

31. Find the ratio of the density of waterat the bottom of a 50.0-m lake to thedensity of water at the surface of thelake. The pressure at the bottom ofthe lake is 4.90 X 105 N/m2.(Hint: the volume of the water willbe decreased by the pressure of thewater above it.) The bulk modulusfor water is 0.21 X 10,0N/m2.

Addi t ional Problems32. A lead block 50.0 cm long, 10.0 cm

wide, and 10.0 cm thick, has a forceof 200,000 N placed on it. Find thestress, the strain, and the change inlength if (a) the block is standingupright, and (b) the block is lyingfiat.

33. An aluminum cylinder must supporta load of 450,000 N. The cylinder is5.00 m high. If the maximumallowable stress is 1.4 X 108, whatmust be the minimum radius of thecylinder in order for the cylinder tosupport the load? What will be thelength of the cylinder when underload?

34. This is essentially the same problemas 33, but now the cylinder is madeof steel. Find the minimum radius ofthe steel cylinder that is necessary tosupport the load and compare it tothe radius of the aluminum cylinder.The maximum allowable stress forsteel is 2.4 X 1010 N/m2.

35. How many 1.00-kg masses may behung from a 1.00-m steel wire, 0.750mm in diameter, without exceedingthe elastic limit of the wire?

36. A solid copper cylinder 1.50 m longand 10.0 cm in diameter, has a massof 5000 kg placed on its top. Find thecompression of the cylinder.

37. This is the same problem as 36,except that the cylinder is an annularcylinder with an inner radius of 3.50cm and outer radius of 5.00 cm. Findthe compression of the cylinder andcompare with problem 36.

38. This is the same problem as problem36 except the body is an I-beam withthe dimensions shown in the diagram.Find the compression of the I-beamand compare to problems 36 and 37.The crossbar width is 2.00 cm.

10 cm"

2 cm

+39. Two pieces of metal rod, 2.00 cmthick, are to be connected togetherby riveting a steel plate to them asshown in the diagram. Two rivets,each 1.00 cm in diameter, are used.What is the maximum force that canbe applied to the metal rod withoutexceeding a shearing stress of 8.4 X10* N/m2.

Rivet Rivet! !

+40. A copper and steel wire are weldedtogether at their ends as shown. Theoriginal length of each wire is 50.0cm and each has a diameter of 0.780mm. A mass of 10.0 kg is suspendedfrom the combined wire. By howmuch will the combined wire stretch?

<-1 Cu

L2 Steel

294 Vibratory Motion, Wave Motion, and Fluids

+41. A copper and steel wire each 50.0 cmin length and 0.780 mm in diameterare connected in parallel to a load of98.0 N, as shown in the diagram. Ifthe strain is the same for each wire,find (a) the force on wire 1, (b) theforce on wire 2, and (c) the totaldisplacement of the load.

+43. Two steel wires of diameters 1.50mm and 1.00 mm, and each 50.0 cmlong, are welded together in series asshown in the diagram. If a weight of98.0 N is suspended from the bottomof the combined wire, by how muchwill the combined wire stretch?

+45. Two springs are connected in seriesas shown in the diagram. The springconstants are kt = 5.00 N/m and k2= 3.00 N/m. A force of 10.0 N isapplied as shown. Find (a) thedisplacement of mass m, (b) thedisplacement of spring 1, and (c) thedisplacement of spring 2.

50.0 cm

50.0 cm

+42. Repeat problem 41 with the diameterof wire 1 equal to 1.00 mm and thediameter of wire 2 equal to 1.50 mm.

+44. Two springs are connected in parallelas shown in the diagram. The springconstants are ki = 5.00 N/m andk2 = 3.00 N/m. A force of 10.0 N isapplied as shown. If the strain is thesame in each spring, find (a) thedisplacement of mass m, (b) theforce on spring 1, and (c) the forceon spring 2.

m

F

X

Interactive TutorialsU 46. Young's modulus for a wire is Y =

2.10 X 10" N/m2. The wire has aninitial length of L0 = 0.700 m and adiameter d = 0.310 mm. A force F= 1.00 N is applied in steps from1.00 to 10.0 N. Calculate the wire'schange in length AL with increasingload F, and graph the result.

* l m

F

k2

x ■

Chapter 10 Elast ici ty 2 9 5