Question:-
For solid spheres falling vertically downwards under gravity in viscous fluid, the terminal, V1 varies with diameter ‘D’ of the sphere as
Option (A)
V1 ∝ D1/2 for all diameters
Option (B)
V1 ∝ D2 for all diameters
Option(C)
V1 ∝ D1/2 for large D and V1 ∝ D2 for small D
Option(D)
V1 ∝ D2 for large D and V1 ∝ D1/2 for small D
Correct Option:
(B)
Question Solution:
Stokes’ formula forms the basis for determination of viscosity of oils which consists of allowing a sphere of known diameter to fail freely in the oil. After initial acceleration. The sphere attains a constant velocity known as terminal velocity which is reached when the external drag on the surface and buoyancy, both acting upwards and in opposite to the motions become equal to the downward force due to gravity
6pmUtR + (4/3) pR3rg = (4/3) pR3rtg
(drag) + (Buoyancy) = (Gravity)
r = density of fluid
rt = density of the sphere material
Ut = terminal velocity
6mUt = (4/3)R2(rt-r)g
Ut = (2R2/9m)(rt-r)g
Ut = (D2/18m) (rt-r)g