Units and Measurement

Unit of temperature is Fahrenheit

Unit of mass is atomic mass of unit

Unit of length is light year

Unit of time is shake

In a measurement, a choice of change of different units does not change the number of significant figures.

In a number without decimal, trailing zeros are insignificant. If a number has a decimal, then preceding zeros are insignificant.

Hence, in 40.40 all zeros are significant

$F=ma=\frac{ML}{T^2}$

$\Rightarrow T^2=\frac{ML}{F}$

$\Rightarrow \left[T\right]=\lfloor M^{1/2}{L}^{1/2}{F}^{-1/2}\rfloor$

(b)  L₁₌  Length of rod =20.123cm

L₂₌  Length of rod = 18.1cm

Difference in length  

= 20.123-18.1

 =2.0 cm

$LC=\frac{1}{100}mm=0.01mm$

Zero error = +0.08 mm.

Diameter = 1 + 72 * 0.01 – 0.08 = 1.64 mm

Radius = 0.82 mm

$T=2\pi \sqrt[]{\frac{L}{g}}or{T}^2=4{\pi }^2\left(\frac{L}{g}\right)$

$\Rightarrow g=4{\pi }^2\left(\frac{L}{T^2}\right)$

$\Rightarrow \frac{\Delta g}{g}\mathrm{\%}=\left(\frac{\Delta L}{L}+\frac{2\Delta T}{T}\right)\mathrm{\%}=\left[\frac{1mm}{1m}+\frac{2*0.01}{1.95}\right]*100\mathrm{\%}=\left(0.001+0.0102\right)*100=1.13\mathrm{\%}$

Since, $\frac{x^2}{\alpha kT}$ $\frac{{}^{}}{}$ should be dimensionless.

So, dimension of $\alpha ,\left[\alpha \right]=\frac{L^2}{M{L}^2{T}^{-2}}=M^{-1}{T}^2$ $\frac{{}^{}}{{}^{}{}^{}}{}^{}{}^{}$

Dimension of $\alpha {\beta }^2$ ${}^{}$ should be that of W.

So, $\left[\alpha {\beta }^2\right]=M{L}^2{T}^{-2}$ ${}^{}{}^{}{}^{}$

$\Rightarrow \left[{\beta }^2\right]=\frac{M{L}^2{T}^{-2}}{M^{-1}{T}^2}=M^2{L}^2{T}^{-4}\Rightarrow \left[\beta \right]=ML{T}^{-2}$

Let 'x' he the value of one division of main scale

              $x=\frac{1}{20}cm=0.05cm$  

              Let y be value of one division on venire scale given

              10 y = 9 x

              $y=\frac{9x}{10}$  

              Least count = $x-\frac{9x}{10}=\frac{x}{10}=\frac{0.05}{10}$  

              = 0.005 cm

              = 5 * 10^-2 mm

Least  count of Vernier = 0.1mm

Reading of Vernier Scale = 5 * 0.1 = 0.5mm

The corrected diameter of sphere = Main Scale Reading + Vernier Scale reading + Zero correction = 1.7 + 0.05 + 0.05 = 1.8cm = 180 * 10^-2 cm.