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55. Problem 13.54P (HRW) A lead brick rests horizontally on cylinders A and
B. The areas of the top faces of cylinders are related by
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;
the Young’s moduli of cylinders are related by
.
The cylinders had identical lengths before the brick was placed on them. What
fraction of the brick’s weight is supported (a) by cylinder A and (b)
by cylinder B? The horizontal distances between the centre of mass of the
brick and the centres of lines of the cylinders are
for cylinder A and
for
cylinder B. (c) What is the ratio
?
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Solution: Click For PDF Version (a) and (b) Let W be the weight of the brick. Let
By definition
As the cylinders had identical lengths before stress was applied and undergo same change in length after being subjected to stress by the weight of the brick, this requirement is equivalent to the condition that strains on the two cylinders are the same. This gives the relation
Using the data given in the problem, the above relation can be rewritten as
Therefore,
(c) Using the results above, we find
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and
be the
fractions of the weight W supported by cylinders A and B,
respectively. As
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