Q28 of 29 Page 70

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.

Let us assume that Tree is uniform, and considering tree is just about to fall.




Let the distance from ground = h/2


Radius of curvature =R


Radius of Tree=r


When Tree will, one part of radius will remain but other will be different.



Using Properties of Triangle we can write,



And we know that d<<R, therefore its square will be less compared, hence neglecting d2, we can write,



It is given in questions elastic torque is given by



It will be equal to,





More from this chapter

All 29 →
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?


 26

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.)

 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 ]

 29

A stone of mass m is tied to an elastic string of negligible mass and spring constant k. The unstretched length of the string is L and has negligible mass. The other end of the string is fixed to a nail at a point P. Initially the stone is at the same level as the point P. The stone is dropped vertically from point P.

(a) Find the distance y from the top when the mass comes to rest for an instant, for the first time.


(b) What is the maximum velocity attained by the stone in this drop?


(c) What shall be the nature of the motion after the stone has reached its lowest point?