Question

A solid spherical bead of lead (uniform density = 11000 kg/m\(^3\)) of diameter \( d = 0.1 \) mm sinks with a constant velocity \( V \) in a large stagnant pool of a liquid (dynamic viscosity = \( 1.1 \times 10^{-3} \) kg·m\(^{-1}\)·s\(^{-1}\)). The coefficient of drag is given by \[ C_D = \frac{24}{Re}, \] where the Reynolds number \( Re \) is defined on the basis of the diameter of the bead. The drag force acting on the bead is expressed as \[ D = (C_D)(0.5 \rho V^2)\left( \frac{\pi d^2}{4} \right), \] where \( \rho \) is the density of the liquid. Neglect the buoyancy force. Using \( g = 10 \) m/s\(^2\), the velocity \( V \) is ________________ m/s.

• A. \( \frac{1}{24} \)
• B. \( \frac{1}{6} \)
• C. \( \frac{1}{18} \)
• D. \( \frac{1}{12} \)

Hint:

To solve for the velocity of an object in a fluid, use the drag force equation with the Reynolds number, and neglect buoyancy for simpler cases.

Answer 1

The Correct Option is C

Step 1: Find the Reynolds number.
The Reynolds number for a spherical object is given by \[ Re = \frac{\rho V d}{\mu}, \] where \( \rho \) is the density of the liquid, \( V \) is the velocity, \( d \) is the diameter of the bead, and \( \mu \) is the dynamic viscosity. Step 2: Apply the drag force formula.
Using the formula for drag force and substituting the values for \( C_D \), \( \rho \), and \( V \), we solve for the velocity \( V \). Final Answer: \[ \boxed{\frac{1}{18}} \]