1.If force $F$, velocity $V$, and time $T$ are taken as fundamental quantities, find the dimensions of Mass and Surface Tension.
Sol: Mass $[M] = [F V^{-1} T^1]$. Surface Tension $[S] = [F / L] = [F / (VT)] = [F V^{-1} T^{-1}]$.
2.A student measures the value of $g$ using a simple pendulum. Length $L = (100 \pm 0.1)\text{ cm}$ and time for 20 oscillations $t = (40 \pm 1)\text{ s}$. Find the maximum percentage error in $g$.
Sol: $g = 4\pi^2 \frac{L}{t^2} \implies \frac{\Delta g}{g}\times 100\% = \frac{\Delta L}{L}\times 100\% + 2\left(\frac{\Delta t}{t}\times 100\%\right) = \frac{0.1}{100}\times 100\% + 2\left(\frac{1}{40}\times 100\%\right) = 0.1\% + 5\% = 5.1\%$.
3.Using Buckingham $\pi$-theorem or dimensional analysis, express the drag force $F$ on a sphere of radius $r$ moving with velocity $v$ in a fluid of density $\rho$ and viscosity $\eta$.
Sol: Stokes' law regime ($Re \ll 1$): $F = 6\pi \eta r v$. For high Reynolds number, $F \propto \rho r^2 v^2$.
