Vardaan Institute - Level 2 Solutions | Units and Measurements

Level 2 Solutions

In a new system of units, speed of light $c$, gravitational constant $G$, and Planck's constant $h$ are taken as fundamental units. Find dimensions of Mass, Length, and Time in this system.

Sol: Mass $M \propto \sqrt{\frac{h c}{G}}$, Length $L \propto \sqrt{\frac{h G}{c^3}}$, Time $T \propto \sqrt{\frac{h G}{c^5}}$ (Planck units).

A physical quantity $P = \frac{a^3 b^2}{\sqrt{c} d}$. If relative errors in $a, b, c, d$ are $1\%, 3\%, 4\%, 2\%$, find maximum percentage error in $P$.

Sol: $\frac{\Delta P}{P}\times 100\% = 3(1\%) + 2(3\%) + \frac{1}{2}(4\%) + 1(2\%) = 3\% + 6\% + 2\% + 2\% = 13\%$.

The main scale of a Vernier Calliper has $1\text{ mm}$ divisions. $10$ divisions of Vernier scale coincide with $9$ main scale divisions. Find its least count.

Sol: $1\text{ VSD} = 0.9\text{ MSD}$. Least Count = $1\text{ MSD} - 1\text{ VSD} = 1.0\text{ mm} - 0.9\text{ mm} = 0.1\text{ mm} = 0.01\text{ cm}$.

A screw gauge has pitch $0.5\text{ mm}$ and $50$ circular divisions. Find least count. If zero mark lies 2 divisions below main line, find zero error.

Sol: Least count = $\frac{0.5\text{ mm}}{50} = 0.01\text{ mm}$. Zero error = $+2 \times 0.01\text{ mm} = +0.02\text{ mm}$ (positive zero error).

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