JEE Advanced 2013 · Paper 1 · Physics Q12

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PhysicsMultiple correct+4 / -1Question 12 of 60

A solid sphere of radius $R$ and density $\rho$ is attached to one end of a mass-less spring of force constant $k$ . The other end of the spring is connected to another solid sphere of radius $R$ and density $3\rho$ . The complete arrangement is placed in a liquid of density $2\rho$ and is allowed to reach equilibrium. The correct statement(s) is (are)

the net elongation of the spring is

\dfrac{4\pi R^3\rho g}{3k}

the net elongation of the spring is

\dfrac{8\pi R^3\rho g}{3k}

the light sphere is partially submerged.

the light sphere is completely submerged.

Answer and worked solution

Official answer: A, D

Can the pair float? Total weight $(\rho+3\rho)Vg=4\rho Vg$ ; the largest possible buoyancy (both fully submerged) is $2\times2\rho Vg=4\rho Vg$ . They balance exactly only when both are fully submerged, so the light sphere is completely submerged.

Spring. For the heavy sphere: $3\rho Vg=2\rho Vg+kx$ , so
$x=\frac{\rho Vg}k=\frac{4\pi R^3\rho g}{3k}.$

(A), (D)

Topics:Buoyancy and Archimedes' principle Spring-mass systems

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JEE Advanced 2013 · Paper 1, Physics question 12