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Question

A man grows into a giant such that his linear dimensions increases by a factor of 9. If his density remains same, then stress on the leg will increase by a factor of
(Assume cubical shape of man with L as edge length).

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Solution

Let the initial mass of man is m and cross-sectional area of his legs is A then mass is given as,
m=ρV=ρL3 and A=L2
[ assuming cubical shape of man having edge length L ]
External load acting on man is his weight (mg)
stress=forcearea=mgA=ρL3gL2
σi=ρLg ....(1)
When man's linear dimensions increases by a factor of 9, Let the final mass of man is m and area of leg is A
m=ρV=ρ(9L)3=93ρL3 and A=(9L)2=92L2
stress=forcearea=mgA=93ρL3g92L2
σ=9ρLg ....(2)
From Eq.(1) and (2),
σσ=9
σ=9σ
Hence stress in the leg will increase by a factor of 9.

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