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Physics · Textbook solutions

Mechanical Properties of Solids

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 23 questions

8.5 Elastic Moduli

5 q

Solved Examples

Worked · 5
  1. Eg 8.1
    A structural steel rod has a radius of 10 mm and a length of 1.0 m. A 100 kN force stretches it along its length. Calculate (a) stress, (b) elongation, and (c) strain on the rod. Young's modulus, of structural steel is 2.0×1011 N m−22.0 \times 10^{11}\,\text{N m}^{-2}.
  2. Eg 8.2
    A copper wire of length 2.2 m and a steel wire of length 1.6 m, both of diameter 3.0 mm, are connected end to end. When stretched by a load, the net elongation is found to be 0.70 mm. Obtain the load applied.
  3. Eg 8.3
    In a human pyramid in a circus, the entire weight of the balanced group is supported by the legs of a performer who is lying on his back (as shown in Fig. 8.4). The combined mass of all the persons performing the act, and the tables, plaques etc. involved is 280 kg. The mass of the performer lying on his back at the bottom of the pyramid is 60 kg. Each thighbone (femur) of this performer has a length of 50 cm and an effective radius of 2.0 cm. Determine the amount by which each thighbone gets compressed under the extra load.
  4. Eg 8.4
    A square lead slab of side 50 cm and thickness 10 cm is subject to a shearing force (on its narrow face) of 9.0×104 N9.0 \times 10^{4}\,\text{N}. The lower edge is riveted to the floor. How much will the upper edge be displaced?
  5. Eg 8.5
    The average depth of Indian Ocean is about 3000 m. Calculate the fractional compression, ΔV/V\Delta V/V, of water at the bottom of the ocean, given that the bulk modulus of water is 2.2×109 N m−22.2 \times 10^{9}\,\text{N m}^{-2}. (Take g=10 m s−2g = 10\,\text{m s}^{-2})

Exercises

18 q
  1. Ex 8.1
    A steel wire of length 4.7 m4.7\,\text{m} and cross-sectional area 3.0×10−5 m23.0 \times 10^{-5}\,\text{m}^2 stretches by the same amount as a copper wire of length 3.5 m3.5\,\text{m} and cross-sectional area of 4.0×10−5 m24.0 \times 10^{-5}\,\text{m}^2 under a given load. What is the ratio of the Young's modulus of steel to that of copper?
  2. Ex 8.2
    Figure 8.9 shows the strain-stress curve for a given material. What are (a) Young's modulus and (b) approximate yield strength for this material?
  3. Ex 8.3(a)
    The stress-strain graphs for materials AA and BB are shown in Fig. 8.10. The graphs are drawn to the same scale. (a) Which of the materials has the greater Young's modulus?
  4. Ex 8.3(b)
    The stress-strain graphs for materials AA and BB are shown in Fig. 8.10. The graphs are drawn to the same scale. (b) Which of the two is the stronger material?
  5. Ex 8.4(a)
    Read the following statement below carefully and state, with reasons, if it is true or false. (a) The Young's modulus of rubber is greater than that of steel;
  6. Ex 8.4(b)
    Read the following statement below carefully and state, with reasons, if it is true or false. (b) The stretching of a coil is determined by its shear modulus.
  7. Ex 8.5
    Two wires of diameter 0.25 cm0.25\,\text{cm}, one made of steel and the other made of brass are loaded as shown in Fig. 8.11. The unloaded length of steel wire is 1.5 m1.5\,\text{m} and that of brass wire is 1.0 m1.0\,\text{m}. Compute the elongations of the steel and the brass wires.
  8. Ex 8.6
    The edge of an aluminium cube is 10 cm10\,\text{cm} long. One face of the cube is firmly fixed to a vertical wall. A mass of 100 kg100\,\text{kg} is then attached to the opposite face of the cube. The shear modulus of aluminium is 25 GPa25\,\text{GPa}. What is the vertical deflection of this face?
  9. Ex 8.7
    Four identical hollow cylindrical columns of mild steel support a big structure of mass 50,000 kg50{,}000\,\text{kg}. The inner and outer radii of each column are 3030 and 60 cm60\,\text{cm} respectively. Assuming the load distribution to be uniform, calculate the compressional strain of each column.
  10. Ex 8.8
    A piece of copper having a rectangular cross-section of 15.2 mm×19.1 mm15.2\,\text{mm} \times 19.1\,\text{mm} is pulled in tension with 44,500 N44{,}500\,\text{N} force, producing only elastic deformation. Calculate the resulting strain?
  11. Ex 8.9
    A steel cable with a radius of 1.5 cm1.5\,\text{cm} supports a chairlift at a ski area. If the maximum stress is not to exceed 108 N m−210^{8}\,\text{N m}^{-2}, what is the maximum load the cable can support?
  12. Ex 8.10
    A rigid bar of mass 15 kg15\,\text{kg} is supported symmetrically by three wires each 2.0 m2.0\,\text{m} long. Those at each end are of copper and the middle one is of iron. Determine the ratios of their diameters if each is to have the same tension.
  13. Ex 8.11
    A 14.5 kg14.5\,\text{kg} mass, fastened to the end of a steel wire of unstretched length 1.0 m1.0\,\text{m}, is whirled in a vertical circle with an angular velocity of 2 rev/s2\,\text{rev/s} at the bottom of the circle. The cross-sectional area of the wire is 0.065 cm20.065\,\text{cm}^2. Calculate the elongation of the wire when the mass is at the lowest point of its path.
  14. Ex 8.12
    Compute the bulk modulus of water from the following data: Initial volume =100.0 litre= 100.0\,\text{litre}, Pressure increase =100.0 atm= 100.0\,\text{atm} (1 atm=1.013×105 Pa)(1\,\text{atm} = 1.013 \times 10^{5}\,\text{Pa}), Final volume =100.5 litre= 100.5\,\text{litre}. Compare the bulk modulus of water with that of air (at constant temperature). Explain in simple terms why the ratio is so large.
  15. Ex 8.13
    What is the density of water at a depth where pressure is 80.0 atm80.0\,\text{atm}, given that its density at the surface is 1.03×103 kg m−31.03 \times 103\,\text{kg m}^{-3}?
  16. Ex 8.14
    Compute the fractional change in volume of a glass slab, when subjected to a hydraulic pressure of 10 atm10\,\text{atm}.
  17. Ex 8.15
    Determine the volume contraction of a solid copper cube, 10 cm10\,\text{cm} on an edge, when subjected to a hydraulic pressure of 7.0×106 Pa7.0 \times 10^{6}\,\text{Pa}.
  18. Ex 8.16
    How much should the pressure on a litre of water be changed to compress it by 0.10%0.10\%?