Two metal spheres of densities in the ratio 3:2 and diameter in the ratio 1:2 are released from rest in two vertical liquid columns of coefficient of viscosity in the ratio 4:3. If the viscous force on them is same, then the ratio of their instantaneous velocities is
Question
Two metal spheres of densities in the ratio 3:2 and diameter in the ratio 1:2 are released from rest in two vertical liquid columns of coefficient of viscosity in the ratio 4:3. If the viscous force on them is same, then the ratio of their instantaneous velocities is
Solution
The viscous force on a sphere moving through a fluid is given by Stokes' Law, which states:
F = 6πηrv
where: F is the viscous force, η is the coefficient of viscosity, r is the radius of the sphere, and v is the velocity of the sphere.
Given that the viscous force on both spheres is the same, we can set up the following equation:
6πη₁r₁v₁ = 6πη₂r₂v₂
We are given that the ratio of the densities is 3:2, the ratio of the diameters (and therefore the radii) is 1:2, and the ratio of the coefficients of viscosity is 4:3. Substituting these ratios into the equation gives:
4v₁ = 3(2)v₂ 4v₁ = 6v₂
Therefore, the ratio of their instantaneous velocities is 4:6 or 2:3.
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