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Q.

A hemispherical bowl of radius R=0.1m is rotating about its own axis (which is vertical) with an angular velocity ω. A particle of mass 10-2 kg on the friction-less inner surface of the bowl is also rotating with the same w. The particle is at height h from the bottom of the bowl. It is desired to measure ‘g’ using the set-up by measuring h accurately. Assuming that R and ω are known precisely and that the least count in the measurement of h is 10-4 m. The minimum possible error (Dg) in the measured value of g is n×102m/s2. Then n is (h << R)

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a

1

b

12

c

3

d

13

answer is A.

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Detailed Solution

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 (a) R is the radius of the bowl​sinθ=rRr=Rsinθ1Ncosθ=mg2andNsinθ=mrω23

Nsinθ=mω2RsinθN=mRω24from 2 & 4mRω2cosθ=mg

g=ω2Rh=ω2R1hR​ taking 'log' both side ​logg=logω2R+log1hRlogg=logω2R+hRh22R2h33R3+

Differentiating with respect to h1gdgdh=01R higher order of hR neglectedΔg=gR×Δh=100.1×104=10×103=1×102

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A hemispherical bowl of radius R=0.1m is rotating about its own axis (which is vertical) with an angular velocity ω. A particle of mass 10-2 kg on the friction-less inner surface of the bowl is also rotating with the same w. The particle is at height h from the bottom of the bowl. It is desired to measure ‘g’ using the set-up by measuring h accurately. Assuming that R and ω are known precisely and that the least count in the measurement of h is 10-4 m. The minimum possible error (Dg) in the measured value of g is n×10−2m/s2. Then n is (h << R)