A student performs an experiment to determine the Young’s modulus of a wire, exactly $$2m$$ long, by Searle’s method. In a particular reading, the student measures the extension in the length of the wire to be $$0.8mm$$ with an uncertainty of ±$$0.05$$ mm at a load of exactly $$1kg$$. The student also measures the diameter of the wire to be $$0.4$$ mm with a uncertainty of ±$$0.01$$ mm. The Young’s modulus obtained from the reading is [Take g $$=$$ $$9.8m/s2$$ exact].

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Infinity Learn · Accepted Answer

$$Y=FLAl=4Mgl$$$$π$$$$d2lM=1.0kg$$ (exact). $$g=9.8ms$$−$$2$$ (exact) $$L=2m$$ (exact) ,$$1=0.8mm=0.8$$×$$10$$−$$3m$$Δ$$l=$$±$$0.05mm$$⋅$$d=0.4mm=0.4$$×$$10$$−$$3m$$Δ$$l=$$±$$0.01mmSubstituting$$ the values of M. g, L, d and I in Eq (i) we $$gety=2.0$$×$$1011Nm$$−$$2From$$ $$Eq(1)$$ the proportionate uncertainty in Y is given byΔ$$YY=$$Δ$$MM+$$Δ$$gg+$$Δ$$LL+2$$Δ$$dd+$$ΔllSince the value of M, g and L are exact, Δ$$M=0$$Δ$$g=0$$ and Δ$$L=0$$, henceΔ$$YY=2$$Δ$$dd+$$Δ$$ll=2$$×$$0.01mm0.4mm+0.05mm0.8mm=0.05+0.0625=0.1125$$Δ$$Y=0.1125$$×$$Y=0.1125$$×$$2.0$$×$$1011=0.225$$×$$1011Nm$$−$$2$$

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