Project ForgeA solid spherical bead of lead (uniform density = 11000 kg/m³) of diameter d=0.1 mm sinks with a constant velocity V in a large stagnant pool of a liquid (dynamic viscosity = 1.1×10⁻³ kg·m⁻¹·s⁻¹). The coefficient of drag is given by c_D = 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)(0.5ρV²)(πd²/4), where ρ is the density of the liquid. Neglect the buoyancy force. Using g=10 m/s², the velocity V is ___________ m/s.
- A1/24
- B1/6
- 1/18
- D1/12
Explanation
Substituting c_D=24μ/(ρVd) into the drag formula gives Stokes' drag: D=3πμVd. At terminal velocity, D equals the bead's weight (buoyancy neglected): 3πμVd = ρ_lead(πd³/6)g, so V=ρ_lead·d²·g/(18μ) = 11000×(1×10⁻⁴)²×10/(18×1.1×10⁻³) = 1.1×10⁻³/0.0198 = 1/18 m/s.
Source: GATE ME 2022 Session 1 (ME-1) (2022)
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