Units & Dimensions

JEE Main level, numerical answer, medium

26 of 149

Question 26 of 149

A ball of radius rr and density ρ\rho dropped through a viscous liquid of density σ\sigma and viscosity η\eta attains its terminal velocity at time tt, given by t=Aρarbηcσdt=A \rho^{a} r^{b} \eta^{\mathrm{c}} \sigma^{d}, where AA is a constant and a,b,ca, b, c and dd are integers. The value of b+ca+d\frac{b+c}{a+d} is ____\_\_\_\_.

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Answer: 1

Using dimensional analysis on t=Aρarbηcσdt = A\rho^a r^b \eta^c \sigma^d:
the dimensions are [T] = [ML−3]a[L]b[ML−1T−1]c[ML−3]d[M L^{-3}]^a [L]^b [M L^{-1} T^{-1}]^c [M L^{-3}]^d.
Expanding: [T] = Ma+c+dL−3a+b−c−3dT−cM^{a+c+d} L^{-3a+b-c-3d} T^{-c}.
Comparing powers: For T: 1=−c1 = -c, giving c=−1c = -1.
For M: 0=a+c+d0 = a + c + d, so a+d=1a + d = 1.
For L: 0=−3a+b−c−3d=−3a+b+1−3d0 = -3a + b - c - 3d = -3a + b + 1 - 3d, giving b=3(a+d)−1=3(1)−1=2b = 3(a+d) - 1 = 3(1) - 1 = 2.
Therefore (b+c)/(a+d)=(2−1)/1=1(b+c)/(a+d) = (2-1)/1 = 1.

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