A spherical ball of radius r and density d is dropped from rest in a viscous fluid having density ρ and coefficient of viscosity η. Then the power of gravitational force acting on the ball at a time t after it is dropped is [xπ d(d − ρ)g²r⁵/(27η)] [1 − exp(−9ηt/(2dr²))]. Find x.
Physics · Mixed Chapter
JEE Mains

Question

A spherical ball of radius r and density d is dropped from rest in a viscous fluid having density ρ and coefficient of viscosity η. Then the power of gravitational force acting on the ball at a time t after it is dropped is [xπ d(d − ρ)g²r⁵/(27η)] [1 − exp(−9ηt/(2dr²))]. Find x.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: For Stokes drag, terminal speed is vₜ=2r²(d−ρ)g/(9η). With the ball’s mass m=(4/3)πr³d, its velocity grows as v(t)=vₜ[1−exp(−t/τ)], where τ=m/(6πηr)=2dr²/(9η). Step 2: Instantaneous gravitational power is P=mgv(t)=[(4/3)πr³ d g][2r²(d−ρ)g/(9η)] [1−exp(−9ηt/(2dr²))]. Step 3: Collect coefficients: P=[8πd(d−ρ)g²r⁵/(27η)] [1−exp(−9ηt/(2dr²))]. Step 4: Comparing coefficients gives x=8.