Q26 of 29 Page 70

A steel rod of length 2l, cross sectional area A and mass M is set rotating in a horizontal plane about an axis passing through the centre. If Y is the Young’s modulus for steel, find the extension in the length of the rod. (Assume the rod is uniform.)



Force acting in this element due to rotating is



Where


df=force on that small element under consideration


dm=mass of element under consideration


ω=angular velocity (which will remain same for all particles)


x=distance from the axis of rotation


Now, Taking mass per unit length



We have taken 2L as we considering the length of half part to be L




Now we know a relation between young modulus and tension force, but since, force is a function of x hence we can only write the equation for a small element of length dx



Here we have represented extension in length as dr, since our original length is dx



Now we need to multiply it by 2 as we were considering only half length of rod,



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24

Consider a long steel bar under a tensile stress due to force’s F acting at the edges along the length of the bar (Fig. 9.5). Consider a plane making an angle θ with the length. What are the tensile and shearing stresses on this plane?


(a) For what angle is the tensile stress a maximum?


(b) For what angle is the shearing stress a maximum?


 25

(a) A steel wire of mass μ per unit length with a circular cross section has a radius of 0.1 cm. The wire is of length 10 m when measured lying horizontal, and hangs from a hook on the wall. A mass of 25 kg is hung from the free end of the wire. Assuming the wire to be uniform and lateral strains << longitudinal strains, find the extension in the length of the wire. The density of steel is 7860 kg m-3(Young’s modules Y=2×1011 Nm-2).

(b) If the yield strength of steel is 2.5×108 Nm-2, what is the maximum weight that can be hung at the lower end of the wire?


 27

An equilateral triangle ABC is formed by two Cu rods AB and BC and one Al rod. It is heated in such a way that temperature of each rod increases by ∆T. Find change in the angle ABC. [Coeff. of linear expansion for Cu is α1 ,Coeff. of linear expansion for Al is α2 ]

 28

In nature, the failure of structural members usually result from large torque because of twisting or bending rather than due to tensile or compressive strains. This process of structural breakdown is called buckling and in cases of tall cylindrical structures like trees, the torque is caused by its own weight bending the structure. Thus the vertical through the centre of gravity does not fall within the base. The elastic torque caused because of this bending about the central axis of the tree is given by . Y is the Young’s modulus, r is the radius of the trunk and R is the radius of curvature of the bent surface along the height of the tree containing the centre of gravity (the neutral surface). Estimate the critical height of a tree for a given radius of the trunk.