Vardaan Institute - Detailed Solutions: Master Sheet (Units & Measurements)

Section A: Dimensional Analysis Solutions

Find the dimensional formula of Universal Gravitational Constant $G$, Planck's constant $h$, and Coefficient of viscosity $\eta$.

Sol: $[G] = [M^{-1} L^3 T^{-2}]$, $[h] = [M L^2 T^{-1}]$, $[\eta] = [M L^{-1} T^{-1}]$.

By method of dimensions, derive the formula for time period $T$ of a simple pendulum.

Sol: $T = k m^a l^b g^c \implies [T] = [M]^a [L]^b [L T^{-2}]^c \implies a=0, b=1/2, c=-1/2 \implies T = k \sqrt{l/g} = 2\pi\sqrt{l/g}$.

Section B: Error Analysis Solutions

The percentage errors in measurement of mass and speed are $2\%$ and $3\%$ respectively. Estimate max percentage error in kinetic energy.

Sol: $K = \frac{1}{2} m v^2 \implies \frac{\Delta K}{K}\% = \frac{\Delta m}{m}\% + 2 \frac{\Delta v}{v}\% = 2\% + 2(3\%) = 8\%$.

A physical quantity $P = \frac{A^3 B^2}{\sqrt{C} D}$. If percentage errors in $A, B, C, D$ are $1\%, 3\%, 4\%, 2\%$, find percentage error in $P$.

Sol: $\frac{\Delta P}{P}\% = 3(1\%) + 2(3\%) + \frac{1}{2}(4\%) + 1(2\%) = 3\% + 6\% + 2\% + 2\% = 13\%$.

Section C: Significant Figures & Vernier/Screw Gauge Solutions

State rules for significant figures and find the number of significant figures in $0.007020\text{ m}^2$ and $6.022 \times 10^{23}$.

Sol: $0.007020$: 4 SFs ($7, 0, 2, 0$). $6.022 \times 10^{23}$: 4 SFs ($6, 0, 2, 2$).

A Vernier Calliper has 10 divisions on Vernier scale matching 9 divisions on Main scale ($1\text{ MSD} = 1\text{ mm}$). Find least count and measured length if main scale reading is $2.8\text{ cm}$ and 4th Vernier division coincides.

Sol: $LC = 1\text{ MSD} - 1\text{ VSD} = 0.1\text{ mm} = 0.01\text{ cm}$. Reading $= MSR + (VSR \times LC) = 2.8\text{ cm} + (4 \times 0.01\text{ cm}) = 2.84\text{ cm}$.

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