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← JEE Advanced 2014 · Paper 2
PhysicsSingle correct+3 / -1Question 9 of 60

JEE Advanced 2014 · Paper 2, Physics question 9

A planet of radius R=110×R=\frac1{10}\times(radius of Earth) has the same mass density as Earth. Scientists dig a well of depth R5\frac R5 on it and lower a wire of the same length and of linear mass density 10−3 kg m−110^{-3}\ \mathrm{kg\,m^{-1}} into it. If the wire is not touching anywhere, the force applied at the top of the wire by a person holding it in place is (take the radius of Earth =6×106=6\times10^6 m and the acceleration due to gravity on Earth is 10 m s−210\ \mathrm{m\,s^{-2}})

Choose one option
Answer and worked solution

Official answer: B

Surface gravity. For equal density, g∝Rg\propto R, so the planet has gs=1 m s−2g_s=1\ \mathrm{m\,s^{-2}} and radius R=6×105R=6\times10^5 m.

Inside the planet. g(r)=gsrRg(r)=g_s\dfrac rR. The wire runs from r=Rr=R down to r=45Rr=\frac45R.

Weight of the wire.
F=∫4R/5RλgsrR dr=λgs2R(R2−1625R2)=0.18 λgsR.F=\int_{4R/5}^R\lambda g_s\frac rR\,dr=\frac{\lambda g_s}{2R}\left(R^2-\tfrac{16}{25}R^2\right)=0.18\,\lambda g_sR.
F=0.18×10−3×1×6×105=108 N.F=0.18\times10^{-3}\times1\times6\times10^5=108\text{ N}.

(B)

Topics:Variation of g
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