A ball of radius and density dropped through a viscous liquid of density and viscosity attains its terminal velocity at time

Question

A ball of radius $r$ and density $ρ$ dropped through a viscous liquid of density $σ$ and viscosity $η$ attains its terminal velocity at time $t$ , given by $t = A ρ^{a} r^{b} η^{c} σ^{d}$ , where $A$ is a constant and $a, b, c$ and $d$ are integers. The value of $\frac{b + c}{a + d}$ is ______. [2026]

Smart Achievers · Accepted Answer

(1)

$T^{1} = {[M L^{- 3}]}^{a} {[L]}^{b} {[M L^{- 1} T^{- 1}]}^{c} {[M L^{- 3}]}^{d}$

$M^{0} L^{0} T^{1} = M^{a + c + d} L^{- 3 a + b - c - 3 d} T^{- c}$

$On comparing$

$a + c + d = 0$ $. . . (i)$

$- 3 a + b - c - 3 d = 0$ $. . . (i i)$

$- c = 1$ $. . . (i i i)$

$From (i) and (iii): a + d = 1$

$From (ii): a + d = \frac{b - c}{3}$

$\frac{b + 1}{3} = 1 \Rightarrow b = 2$

$b + c = 2 + (- 1) = 1$

$\Rightarrow \frac{b + c}{a + d} = \frac{1}{1} = 1$

Solution

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Solution

Q. A ball of radius r and density ρ dropped through a viscous liquid of density σ and viscosity η attains its terminal velocity at time t, given by t=Aρarbηcσd, where A is a constant and a,b,c and d are integers. The value of b+ca+d is ______. [2026]

Ans. (1) T1=[ML-3]a[L]b[ML-1T-1]c[ML-3]d M0L0T1=Ma+c+dL-3a+b-c-3dT-c On comparing a+c+d=0 ...(i) -3a+b-c-3d=0 ...(ii) -c=1 ...(iii) From (i) and (iii): a+d=1 From (ii): a+d=b-c3 b+13=1⇒b=2 b+c=2+(-1)=1 ⇒b+ca+d=11=1

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