KVPY 2019 SB SX Exam Previous Papers

Question : 100 of 120

Marks: +1, -0

A bubble of radius R in water of density ρ is expanding uniformly at speed v. Given that water is incompressible, the kinetic energy of water being pushed is :

Zero
$2π ρR ^3v ^2$
$2π ρR^3v ^2 \/3$
$4π ρR^3v ^2 \/3$

Validate

Solution:

A_1v_1 = A_2v_2

(4πR^2 )v = (4πx ^2 )v_x

v_x=R^2/x^2.v

so small kinetic energy at x for width

dx = dk =1/2dm(v_x^2)

dk = 1/2[ ρ(4πx^2dx)v_x^2]

=1/2 ρ4πx^2(R^2/x^2v)^2dx

Total kinetic energy
k =

∫dk=2ρπR^4v^2∫↖{∞}↙{R}{dx}/x^2

k=2ρπR^4v^2[-1/x]_R^∞=2ρπR^3v^2

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