KVPY SA Exam 2015 Question Paper

Question : 70 of 80

Marks: +1, -0

Stoke’s law states that the viscous drag force F experienced by a sphere of radius a, moving with a speed v through a fluid with coefficient of viscosity η , is given by

F = 6π η av

If this fluid is flowing through a cylindrical pipe of radius r, length l and a pressure difference of P across its two ends, then the volume of water V which flows through the pipe in time t can be written as

V/t=k(P/l)^a η ^br^c,

where k is a dimensionless constant. Correct values ofa, b and c are

$a = 1, b = - 1, c = 4$
$a = -1, b = 1, c = 4$
$a = 2, b = -1, c = 3$
Insufficient data

Validate

Solution:

V/t=k(P/l)^a η ^br^c,

V → volume
P → pressure
η → coefficient of viscosity
r → radius
Using dimensional analysis

[M^0 L ^3 T ^{–1} ] = [M^1 L ^{-2} T ^{-2} ] ^a [M^1 L ^{-1} T ^{-1} ] ^b [L]^c

[M^0 L ^3 T ^{-1} ] = [M^{a+b} L^{-2a-b+c} T^{-2a-b} ]

a + b = 0 ...(1)
–2a – b + c = 3 ...(2)
a = –b
–2a – b = – 1 ...(3)
put the value of – 2a – b = –1 in equation (2)
–1 + c = 3
c = 4
put a = – b in equation (3)
2b – b = –1 ⇒ b = –1
and a = 1
hence option (A) is correct.

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