A ball of radius r and density dropped through a viscous liquid of density and viscosity attains its terminal velocity at time t, given by t = A ?a r b c d where A is a constant and a,b,c and d are integers. The value of b + c a + d i s _______

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)

$T = {(M L^{- 3})}^{a} L^{b} {(M L^{- 1} T^{- 1})}^{c} (M L^{- 3})^{d}$

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

$on comparing$

$c = - 1; a + c + d = 0; - 3 a - c - 3 d + b = 0$

$b = 2; a + d = 1$

$b + c = 1$

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 $t = A ρ^{a} r^{b} η^{c} σ^{d}$ where A is a constant and a,b,c and d are integers. The value of $\frac{b + c}{a + d} is$ _______. [2026]

Ans.
(1)

$T = {(M L^{- 3})}^{a} L^{b} {(M L^{- 1} T^{- 1})}^{c} (M L^{- 3})^{d}$

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

$on comparing$

$c = - 1; a + c + d = 0; - 3 a - c - 3 d + b = 0$

$b = 2; a + d = 1$

$b + c = 1$

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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=A ρarbηcσd where A is a constant and a,b,c and d are integers. The value of b+ca+d is _______. [2026]

Ans. (1) T=(ML-3)aLb(ML-1T-1)c(ML-3)d T=Ma+c+dL-3a-c-3d+bT-c on comparing c=-1; a+c+d=0; -3a-c-3d+b=0 b=2; a+d=1 b+c=1

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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 $t = A ρ^{a} r^{b} η^{c} σ^{d}$ where A is a constant and a,b,c and d are integers. The value of $\frac{b + c}{a + d} is$ _______. [2026]

Ans.
(1)

$T = {(M L^{- 3})}^{a} L^{b} {(M L^{- 1} T^{- 1})}^{c} (M L^{- 3})^{d}$

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

$on comparing$

$c = - 1; a + c + d = 0; - 3 a - c - 3 d + b = 0$

$b = 2; a + d = 1$

$b + c = 1$