# 9 MECHANICAL PROPERTIES OF SOLIDS

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1 9 MECHANICAL PROPERTIES OF SOLIDS Deforming force Deforming force is the force which changes the shape or size of a body. Restoring force Restoring force is the internal force developed inside the body which brings the body back to original shape and size when a deforming force acts on it. Elasticity It is the property of a body by virtue of which it tends to regain its original shape and size when deforming force is removed. Difference between perfectly elastic bodies and plastic bodies If a body immediately regains its original shape and size when the applied force is removed, it is perfectly elastic. eg: quartz If a body does not show any tendency to regain its original shape and size when the applied force is removed, it is a plastic body. eg: putty, mud etc. Stress It is the restoring force acting per unit area. Stress. Its unit is Strain Strain Strain has no unit. Longitudinal stress or linear stress It is the stress when there is change in length. Tensile stress causes increase in length. Compressive stress causes decrease in length. Normal stress or hydraulic stress It is the stress when there is change in volume. Shearing stress or tangential stress Shearing stress or tangential stress is the stress when there is change in shape of the body. Linear strain Linear strain is the ratio of change in length to the original length. Volume strain Volume strain is the ratio of change in the volume to the original volume. The strain produced by hydraulic stress is called volume strain. Shearing strain Shearing strain indicates the angle through which deformation take place. Unique Learning Centre, Ulloor, Tvpm. Phone: Page 1

2 The area of ropes used in cranes Shearing strain tan where L is the height and is the tangential displacement. Hooke s law Area of cross section of the rope where W is weight attached and is the yield strength (For steel, yield strength ) (Here W mg). Ideas to be kept during the construction of a beam for building For small deformations, the stress is directly proportional to the strain. Stress strain or the modulus of elasticity. constant. This constant is called Modulus of elasticity There are three moduli of elasticity. They are Young s modulus, Bulk modulus and Shear modulus (rigidity modulus). Unit of modulus of elasticity: Young s modulus It is the ratio of linear stress to linear strain. Young s modulus Bulk modulus It is the ratio of normal stress to volume strain. Bulk modulus Bulk modulus Shear modulus Shear modulus Shear modulus Young s modulus of the material of a wire Young s modulus where M is the mass attached to wire, is the length of wire, s its radius and is the extension in the wire. Compressibility Reciprocal of bulk modulus is called compressibility. The sag ( ) produced in the beam when weight is placed over it, where W is the weight, is the length of the beam, b is the breadth of the beam, d is the depth of the beam and Y is the young s modulus of the material of the beam. Cross sectional shape is ideal for load bearing bars This shape provides maximum load bearing surface. As the depth of it is more, the sag produced in it will be less. As it requires less material, cost can be reduced. Weight of beam is minimum as the material required is minimum. At the same time it provides maximum strength. The maximum height of a mountain on the earth s surface is 10km The stress per area must be less than elastic limit of the rocks at the base of the mountain. The elastic limit of rock is h g Stress-strain graph for a metal In the first section, the stress and strain are proportion up to a limit called proportional limit (point A in figure). Unique Learning Centre, Ulloor, Tvpm. Phone: Page 2

3 The body behaves as elastic. In the section from A to B the stress and the strain are not in direct proportion. But if force is removed, the body returns to its original position. The point B is called elastic limit or yield point. The corresponding stress is called yield strength. Poisson s Ratio If the length of a wire changes from to and diameter changes from to, Poisson s ratio / / Poisson s ratio has no unit or dimension... Elastic fatigue When a material is subjected to repeated stresses over a long period,it loses its strength. This is known as elastic fatigue. Elastic strain energy From B onwards if stress is increased, the strain increases much up to the point D. When stress is removed from B to D ( at C ) the body will not regain its original shape and size. It attains permanent set. The deformation is called plastic deformation. The point D is called ultimate yield point. Beyond it additional strain is produced even for decreased stress and the body breaks at the point E called fracture point (breaking point). If D and E are close, the material is brittle. If they are far, it is ductile. Substances which can be extended to produce large values of strain are called elastomers. eg: rubber, elastic tissue of aorta etc. The stress- strain graph for the elastic tissue of aorta Here the elastic region is large. But it does not obey Hooke s law. For causing deformation on a body, work has to be done. This work is stored in the form of elastic strain Expression for elastic strain energy per unit volume of a wire Elastic potential energy stored in the wire (Strain energy) stressstrainvolume Relation among different elastic constants: Y 3B (1-2) Y 2 (1+ ) + ratio, Cantilever where Y Young s modulus, Poisson s B bulk modulus, shear modulus. A beam clamped at one end and loaded at free end is called cantilever. Depression at the free end of a cantilever where w load, l length of the cantilever, Y Young s modulus of elasticity, and geometrical moment of inertia. For a beam of rectangular cross-section having breadth b and thicknes d. Unique Learning Centre, Ulloor, Tvpm. Phone: Page 3

4 For a beam of circular cross-section area having radius r, Multiple choice questions: Class Work 1. The nature of molecular forces resembles with the nature of the (a) gravitational force (b) nuclear force (c) electromagnetic force (d) weak force 2. The potential energy between two molecules as a function of the distance between them has been shown in the figure. The two molecules are (a) Attracted when lies between A and B and are repelled when lies between B and C (b) Attracted when lies between B and C are repelled when lies between A and B (c) Attracted when they reach B (d) Repelled when they reach B 3. Elasticity is due to (a) decrease of with separation between atoms / molecules (b) increase of with separation between atoms/ molecules (c) asymmetric nature of curve (d) None of the above 4. A uniform bar of square cross-section is lying along a frictionless horizontal surface. A horizontal force is applied to pull it from one of it ends, then (a) the bar is under same stress throughout its length (b) the bar is not under any stress because force has been applied only at one end (c) the bar simply moves without any stress in it (d) the stress developed gradually reduces to zero at the end of the bar where no force is applied 5. and are two wires. The radius of is twice that of. They are stretched by the same load. Then, the stress on is (a) equal to that on A (b) four times that on A (c) two times that on A (d) half that on A 6. On suspending a weight Mg, the length of elastic wire having area of cross- section, becomes double the initial length. The instantaneous stress action on the wire is (a) / (c) 2/ (b) / 2 (d) 4/ 7. A 2 long rod is suspended with the help of two wires of equal length. One wire is of steel and its cross-sectional area is 0.1 and another wire is of brass and its crosssectional area is 0.2. If a load is suspended from the rod and stress produced in both the wires is same, then the ratio of tensions in them will be (a) depend on the position of (b) / 2 (c) / 1 (d) / The length of a wire increases by 1% by a load of 2. The linear strain produced in the wire will be (a) 0.02 (b) (c) 0.01 (d) A uniform cube is subjected to volume compression. If each side is decreased by 1%, then bulk strain is (a) 0.01 (b) 0.06 (c) 0.02 (d) A cube of aluminum of side 0.1 m is subjected to a shearing force of 100. The top face of the cube is displaced through 0.2 cm with respect to the bottom face. The shearing strain would be Unique Learning Centre, Ulloor, Tvpm. Phone: Page 4

5 (a) 0.02 (b) 0.1 (c) (d) A copper and a steel wire of the same diameter are connected end to end. A deforming force is applied to this composite wire which causes a total elongation of 1cm. The two wires will have (a) the same stress and strain (b) the same stress but different strain (c) the same strain but different stress (d) different strains and stress 12. A steel rod of length 1 m and radius 10 mm is stretched by a force 100 along its length. The stress produced in the rod is (Given : 2 10 ) (a) (b) (b) (d) The graph shows the behavior of a length of wire in the region for which the substance obeys Hooke s law. and represent The force required to stretch by 0.1 % of its length is (a) 360 (c) For a perfectly rigid body, (b) 36 (d) (a) Young s modulus is infinite and bulk modulus is zero. (b) Young s modulus is zero and bulk modulus is infinite (c) Young s modulus is infinite and bulk modulus is also infinite. (d) Young s modulus is zero and bulk modulus is also zero. 18. If the ratio of diameters, lengths and Young s moduli of steel and brass wires shown in the figure are, and, respectively. Then the corresponding ratio of increase in their lengths would be (a) P applied force, Q extension (b) P extension, Q applied force (c) P extension, Q stored elastic energy (d) P stored elastic energy, Q extension 14. On applying a stress of 20 10, the length of a perfectly elastic wire is doubled. Its Young s modulus will be (a) (c) (b) (d) A wire of length 2 m is made from 10 of copper. A force is applied so that its length increases by 2. Another wire of length 8 is made from the same volume of copper. If the force is applied to it its length will increase by (a) 0.8 cm (b) 1.6 cm (c) 2.4 cm (d) 3.2 cm 16. The diameter of a brass rod is 4 mm and Young s modulus of brass is Unique Learning Centre, Ulloor, Tvpm. Phone: Page 5 (a) (b) (c) (d) 19. Which of the following statements is incorrect? (a) Young s modulus and shear modulus are relevant only for solids. (b) Bulk modulus is relevant for liquid and gases. (c) Metals have larger values of Young s modulus than elastomers. (d) Alloys have larger values of Young s modulus than metals. 20. Identical springs of steel and copper ( are equally stretched. (a) Less work is done on copper spring. (b) Less work is done on steel spring. (c) Equal work is done on both the springs. (d) Data is incomplete. 21. In plotting stress versus strain curves for two materials P and Q, a student by mistake puts strain on the Y-axis and stress on the x- axis

6 as shown in the figure. Then, the correct statement is /are 26. When a pressure of 100 atmosphere is applied on a spherical ball of rubber, then its volume reduces to 0.01%. The bulk modulus of the material of the rubber in dyne is (a) (d) 1 10 (b) (d) (a) has more tensile strength than Q (b) is more ductile than Q (c) is more brittle than Q (d) The Young s modulus of is more than that of Q 22. The Young s modulus of steel is twice that of brass. Two wires of same length and of same area of cross-section, one of steel and another of brass are suspended from the same roof. If we want the lower ends of the wires to be at the same level, then the weight added to the steel and brass wires must be in the ratio of (a) 1 : 2 (b) 2 : 1 (c) 4 : 1 (d) 1 : Two wires and of same material. Their lengths are in the ratio 1 : 2 and diameters are in the ratio 2 : 1 when stretched by force and respectively they get equal increase in their lengths. Then, the ratio / should be (a) 1 : 2 (b) 1 : 1 (c) 2 : 1 (d) 8 : One end of a horizontal thick copper wire of length 2 and radius 2 is welded to an end of another horizontal thin copper wire of length and radius. When the arrangement is stretched by applying forces at two ends, the ratio of the elongation in the thin wire to that in the thick wire is (a) 0.25 (b) 0.50 (c) 2 (d) The upper end of a wire of radius 4 mm and length 100 cm is clamped and its other end is twisted through an angle of 30. Then, angle of shear is 27. A solid sphere of radius made of a material of bulk modulus is surrounded by a liquid in a cylindrical container. A massless piston of area floats on the surface of the liquid. When a mass is placed on the piston to compress the liquid, the fractional change in the radius of the sphere is (a) (b) (c) (d) 28. The edge of an aluminium cube is 10 cm long. One face of the cube is firmly fixed to a vertical wall. A mass of 100 kg is then attached to the opposite face of the cube. The vertical deflection of this face is (Shear modulus of aluminium 25, g 10 ) (a) 4 10 (c) 4 10 (b) 4 10 (d) The approximate depth of an ocean is The compressibility of water is and density of water is 10 /. What fractional compression of water will be obtained at the bottom of the ocean? (a) (c) (b) (d) To what depth must a rubber ball be taken in deep sea,so that its volume is decreased by 0.1 %. (Take, density of sea water as 10 bulk modulus of rubber as 9 10, g 10 ) (a) 9 m (b) 18 m (c) 90 m (d) 180 m 31. Two wires of the same material and length but diameter in the ratio 1:2 are stretched by the (a) 12 (b) 0.12 (c) 1. 2 (d) Unique Learning Centre, Ulloor, Tvpm. Phone: Page 6

7 same load. The ratio of elastic potential energy per unit volume for the two wires is (a) (c) (b) (d) 3 10 (a) 1:1 (b) 2:1 (c) 4:1 (d) 16: When the load on a wire is increased from 3 to 5, the elongation increases from 0.61 mm to 1.02 mm. The required work done during the extension of the wire,is 38. A uniform rod of length and density is being pulled along a smooth floor with a horizontal acceleration. The magnitude of the stress at the transverse cross- section through the mid-point of the road is (a) (c) (b) 8 10 (d) If the work done in stretching a wire by 1 mm is 2, the work necessary for stretching another wire of same material but with double radius of cross section and half the length by 1mm is (a) 16 (b) 8 (c) 4 (d) 1/4 34. A stone of mass is tied to one end of a wire of length. The diameter of the wire is and it is suspended vertically. The stone is now rotated in a horizontal plane and makes an angle with the vertical. If young s modulus of the wire is, then the increase in the length of the wire is (a) (c) (b) (d) 35. To break a wire, a force of 10 is required. If the density of the material is 3 10, then the length of the wire which will break by its own weight will be (a) 34 m (b) 30 m (c) 31 m (d) 29 m 36. A wire of diameter 1 mm breaks under a tension of Another wire of same material as that of the first one, but of diameter 2 mm breaks under a tension of (a) 500 (b) 1000 (c) (d) In steel, the Young s modulus and the strain at the breaking point are 2 10 and 0.15, respectively. The stress at the breaking point for steel is therefore (a) (c) (b) (d) None of these 39. A metal wire of length and area of crosssection is attached to a rigid support. Another metal wire of length and of the same cross- sectional area is attached to the free end of the first wire. A body of mass is then suspended from the free end of the second wire. If and are the Young s moduli of the wires respectively, the effective force constant of the system of two wires is (a) (c) (b) / (d) / / 40. A uniform rod of mass m, length L, area of cross-section A is rotated about an axis passing through one of its ends perpendicular to its length with constant angular velocity in a horizontal plane. If Y is Young s modulus of the material of rod, the increase in its length due to rotation of rod is (a) (b) (c) (d) 41. Two strips of metal are riveted together at their ends by four rivets, each of diameter 6mm. Assume that each rivet is to carry one quarter of the load. If the shearing stress on the rivet is not to exceed , the maximum tension that can be exerted by the riveted strip is Unique Learning Centre, Ulloor, Tvpm. Phone: Page 7

8 (a) 2 10 (b) Statement (c) (d) The lifting capacity of a crane is 10 metric tonne. How thick should the rope steel rope be? (Yield strength of steel , g 10 ) (a) 3 m (b) 1mm (c) 1cm (d) 0.1 cm 43. The Poisson s ratio of a material is 0.4. If a force is applied to a wire of this material, there is a decrease of cross- sectional area 2% The percentage increases in its length is Elongation produced in a body is perpendicular directly proportional to the applied force. Statement This is the law of elasticity, now called as Hooke s law. 47. The graph is drawn between the applied force and the strain () for a thin uniform wire. Now, pick correct statement : (a) 3% (b) 2.5 % (c) 1% (d) 0.5% 44. A uniform rod of length (L) and area of crosssection (A) is subjected to tensile load (F). If be the Poisson s ratio and Y be Young s modulus of the material of the rod, then find the volumetric strain produced in the rod. I. Wire breaks at c. II. Wire becomes hard at a. III. Wire tends to flow at b. IV. Wire becomes plastic at a. (a) (c) zero Directions 1 2 (b) 1 2 (d) None of these (a) Only I (b) Only II (c) Only III (d) Only IV 48. From stress - strain curves for the materials A and B, Choose correct statement. (Q.Nos ) In the following questions, a statement I is followed by a corresponding statement II. Of the following statements, choose the correct one (a) Both Statement and Statement II are correct and Statement is the correct explanation of Statement. (b) Both Statement and Statement are correct but Statement is not the correct explanation of Statement (c) Statement is correct but Statement is incorrect. (d) Statement I is incorrect but Statement is correct. I. A is brittle material II. B is brittle material III. A is ductile material IV. B is ductile material (a) I,II (b) II, III (c) III, IV (d) IV & I 49. The diagram shows a force extension graph for a rubber band. Consider the following statements. 45. Statement When a solid sphere is placed in the fluid under high pressure, then it is compressed uniformly on all sides. Statement The force applied by fluids acts in perpendicular direction at each point of surface. I. It will be easier to compress this rubber than expand it. II. Rubber does not return to its original length after it is stretched. Unique Learning Centre, Ulloor, Tvpm. Phone: Page 8

9 III. The rubber band will get heated if it is stretched and released. Which of these can be deduced from the graph? (a) & (c) & Directions (b) & (d) None of these (Q.Nos ) These questions are based on the following situation. Choose the correct options from those given below. A bar of cross section A is subjected to equal and opposite tensile forces at its ends. Consider a plane section of the bar whose normal makes an angle with the axis of the bar. (b) If A and B are made for two different wires of equal length and of same material, then wire A is more thicker than B. (c) If A and B are drawn for same wire at two different temperatures, then. (d) If A and B are drawn for wires of same material and same area but of different length, then. 55. Two blocks are tied with a wire and are hung over a pulley as shown. Masses of blocks are 1 kg and 4kg and pulley is frictionless. Then, choose correct options. 50. What is the tensile stress on this plane? (a) (/ ) (c) (/ (b) / (d) (/ 51. What is the shearing stress on this plane? (a) sin 2 (b) (c) (d) cos For what value of is the tensile stress maximum? (a) Tension in wire is 10 N (b) Tension in wire is 16 N (c) If breaking stress is Then wire must be of 4 10 m radius so that wire does not break. (d) Acceleration of blocks is more than g. 56. Wire A and B are connected with mass as shown in figure. Wires are of same material and have radii and. End of B pulled with a force of mg/3. (a) 0 (b) 90 (c) 45 (d) For what value of is the shearing stress maximum? (a) 45 (b) 30 (c) 90 (d) Stress - strain graphs A and B are prepared by a student but he forgets to specify what are the conditions for A and B (a) A breaks before B when (b) A breaks before B when < 2 (c) Either A or B may break if 2 (d) Data not sufficient to reach any conclusion (a) If A and B are for two different wires of identical dimensions, then : 3 : Three blocks are connected with wires A and B of same cross-section area and Young s modulus Y. All three block are of mass m each. Unique Learning Centre, Ulloor, Tvpm. Phone: Page 9

10 The instantaneous stress / (a) Tension in wire A (b) Tension in wire B (c) Stress in wire A (d) Strain in wire B Hints and Explanations 1. (c) Inter - molecular and inter - atomic forces are due to electric and magnetic interactions between atoms and molecules. 2. (b) - of slope of U versus graph. or repulsive when is less than equilibrium separation, i.e., between A and B and or attractive when is between B and C. 3. (c) Elasticity occurs to asymmetric nature of U versus graph. When separation of molecules is less than (or more than) equilibrium separation, PE of system increases and system dissipates this energy to reach minimum energy configuration. As a result separation is again restored to equilibrium separation for which U is minimum. 4. (d) Force causes an acceleration and each section of rod experiences a tension which is zero at the other end. 5. (b) Stress Stress 2 4 ( F constant) 6. (c) When the length of wire becomes double, its area of cross- section will become half because volume of wire is constant. 7. (d) Stress 8. (c) Strain 0.01 constant % / 9. (d) Volume of cube, Percentage change in 3 (percentage in L) 3 (1%) 3 % 3% of V Volumetric strain 10. (d) 11. (b) Shearing strain, Stress As each wire is of same diameter, hence the area of cross- section is same for both the wires. As both the wires have the same force (F) and same area of cross- section (A) therefore, stress is same for both the wires. Y or Strain As Y is different for both the wires and stress is same for both the wires, therefore strain is different for both the wires 12. (c) 10 mm m 10 1, Stress produced in the rod is (c) Unique Learning Centre, Ulloor, Tvpm. Phone: Page 10

11 The graph between applied force and extension will be straight line because in elastic range. Applied force extension, But the graph between extension and stored elastic energy will be parabolic nature. U or U 14. (b) Young s modulus As the length of wire gets doubled therefore strain 1 Y Stress (d) Length, ( As, and are constants) mm 32 mm 3.2 cm 16. (a) r 2 10, Y 9 10, 0.1 % Young s modulus, Y F YA (c) 18. (b) 3mg q, P, r (q) 19. (d) Metals have large values of young s modulus than alloys and elastomers. 20. (b) W, and are constants From. (i) and (ii) < < (i) (ii) Less work is done on steel spring. 21. (b) From graph, P has more strain so P is more ductile with respect to Q. 22. (b) 2 and and, Weight added to the steel and brass wires must be in the ratio of 2:1 Young s modulus, Y (2m + 3m) g 5mg 23. (d) 24. (c) Y 1 1 Unique Learning Centre, Ulloor, Tvpm. Phone: Page 11

12 25. (b) r 26. (c) % of / 27. (c) / / / / (c) / Shear modulus, / (100kg) N (10cm 10) pa Substituting the values, (c) d 2700 m, 10 / Compressibility per pascal The pressure at the bottom of ocean is given by h Fractional compression Compressibility (c) h be the depth at which the rubber ball be taken. h Definition of bulk modulus, B / The negative sign shown that with increases in pressure, a decrease in volume occurs. B or h Substituting the values h. 90 m 31. (d) Elastic potential energy per unit volume is / / As both the wires are stretched by the same load, therefore 32. (a) Work done in stretching the wire through 0.61 mm under the load of (a) stretching force extension Work done in stretching the wire through 1.02 mm under the load of The work done in stretching the wire from 0.61 mm to 1.02 mm. - ( ) Stretching force, Unique Learning Centre, Ulloor, Tvpm. Phone: Page 12

13 Both the wire s are of same material, so Y will be equal, extension in both the wires is same, so will be equal. 8 / 8 Work done in stretching a wire, W For same extension, (c) The situation is as shown in the figure Stress strain (b) The force at the cross - section through the mid - point of the rod is 39. (c) When the two wires are connected together in series, the effective force constant is given by Substituting the corresponding lengths, area of cross section and the young s moduli For vertical equilibrium of stone, mg or / (i) 40. (c) Consider a small element of length dx at a distance x from the axis of rotation. Mass of the element, 35. (a) Length of the wire which will break by its own weight 33.3m 34 m. 36. (d) Breaking force area of cross section 37. (d) The centripetal force acting on the element is As this force provided by tension in the rod (due to elasticity) So the tension in the rod at a distance x from the axis of rotation will be due to the centripetal force due to all elements between x to x Unique Learning Centre, Ulloor, Tvpm. Phone: Page 13

14 be increase in length of the element. 41. (c) / / The total elongation of the whole rod is dx ---- (i) On differenting both sides / 100 2% 2 2. / / 2. or % Radius of a rivet, r mm 3mm Maximum load on a rivet Maximum stress Area of cross section Maximum tension that can be exerted by rivet strip (c) M 10 metric tonne If r be the radius of the rope used in the crane, Yield strength r /. r cm 44. (b) Percentage increase in its length, i.e., % 2.5% Young s modulus,. 2 + (1 2 ) / / / / (i) (b) Poisson s ratio, 0.4 / Area, A Or 45. (a) A solid sphere placed in the fluid under high pressure is compressed uniformly on all sides. Unique Learning Centre, Ulloor, Tvpm. Phone: Page 14

15 . Slope of load extension graph As wires are identical, The force applied by the fluid acts in perpendicular direction at each point of the surface and the body is said to be under hydraulic compression. 46. (a) 47. (c) Statement III is correct b is yield point. Wire tends to flow at b. 48. (b) II and III B is for brittle as there is no plastic region. is a ductile material. 49. (b) II and III 50. (a) 51. (a) The resolved part of F along the normal is the tensile force on this plane and the resolved part parallel to the plane is the shearing force on the plane. Tensile stress Area of plane section sec Shearing stress sin cos sin (a) Tensile stress will be maximum when is maximum, i.e., cos 1 or (a) Shearing stress will be maximum when sin 2 is maximum, i.e., sin 2 1 or 2 90, i.e., (a, b) If A and B are prepared for identical wires of two different materials, then : 3 : 1 Extension for wire A is less. A is thicker. 55. (b,c) g 16 N Breaking force Breaking stress Area r 56. (a,b,c) Stress in Stress in Stress in B and stress in Stress in A > Stress in B Wire A will break earlier.if 2, then stress in A stress in B it means either A or B may breaks. If < 2, then stress in A is more than that of B. A may break earlier. 57. (b,d) Tension in B Stress in B.. g Strain in wire B Unique Learning Centre, Ulloor, Tvpm. Phone: Page 15

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