Question

A sphere of density ρ & mass m is moving down with constant velocity in viscous liquid of density ρ₀. Find out viscous force on sphere

• A. \(mg(\frac{1-\rho _{0}}{\rho})\)
• B. \(2mg(\frac{1-\rho _{0}}{\rho})\)
• C. \(mg(\frac{1-3\rho _{0}}{\rho})\)
• D. \(mg(\frac{1+\rho _{0}}{\rho})\)

Answer 1

The Correct Option is A

To solve this problem, we need to determine the viscous force acting on a sphere descending through a viscous liquid with constant velocity. Here are the steps and reasoning used to reach the solution:

1. The problem states that the sphere is moving with a constant velocity, indicating that the net force acting on it is zero due to equilibrium. This highlights a balance between the gravitational force, buoyant force, and viscous force acting on the sphere.
2. The gravitational force acting on the sphere is given by: \(F_g = mg\), where m is the mass of the sphere and g is the acceleration due to gravity.
3. The buoyant force on the sphere, according to Archimedes' principle, is: \(F_b = V \rho_0 g\), where V is the volume of the sphere, and \(\rho_0\) is the density of the liquid.
4. The condition for constant velocity (net force zero) can be expressed as: \(F_g = F_b + F_v\), where F_v is the viscous force. Rearranging gives: \(F_v = F_g - F_b\).
5. Substituting the expressions for F_g and F_b: \(F_v = mg - V \rho_0 g\).
6. Knowing that the volume V of the sphere can be expressed in terms of its density \(\rho\) and mass m as: \(V = \frac{m}{\rho}\), we substitute in the expression above: \(F_v = mg - \left(\frac{m}{\rho}\right)\rho_0 g\).
7. Simplifying, we get: \(F_v = mg\left(1 - \frac{\rho_0}{\rho}\right)\).

Therefore, the viscous force on the sphere is \(mg\left(\frac{1 - \rho_0}{\rho}\right)\), which corresponds with the correct answer option given:

\(mg\left(\frac{1 - \rho_0}{\rho}\right)\).

This option is correct as it correctly balances the forces according to the given conditions of constant velocity (net force being zero).