Units & Measurement and Dimensions

To determine the correct value for $ \sqrt{AB} $, we start with the given measurements:

$ A = 1.0 \, \text{m} \pm 0.2 \, \text{m} $

$ B = 2.0 \, \text{m} \pm 0.2 \, \text{m} $

First, calculate $ Y = \sqrt{AB} $:

$ Y = \sqrt{(1.0)(2.0)} = \sqrt{2.0} \approx 1.414 \, \text{m} $

Rounding this to two significant digits:

$ Y = 1.4 \, \text{m} $

Next, calculate the uncertainty in $ Y $:

The relative uncertainty in $ Y $ is given by:

$ \frac{\Delta Y}{Y} = \frac{1}{2} \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right) $

Substitute the values:

$ \frac{\Delta Y}{Y} = \frac{1}{2} \left( \frac{0.2}{1.0} + \frac{0.2}{2.0} \right) = \frac{1}{2} \left( 0.2 + 0.1 \right) = \frac{0.3}{2} $

Calculate $ \Delta Y $:

$ \Delta Y = \frac{0.3 \times 1.4}{2} = 0.21 $

Rounding off to one significant digit, $\Delta Y \approx 0.2 \, \text{m}$.

Thus, the correct value for $ \sqrt{AB} $ is:

$ 1.4 \, \text{m} \pm 0.2 \, \text{m} $

(n + 1) divisions of vernier scale = n division of main scale

$\therefore$ 1 VSD = ${n \over {n + 1}}$ MSD

Least count = 1 MSD $-$ 1 VSD

= 1 MSD $-$ ${n \over {n + 1}}$ MSD = ${1 \over {n + 1}}$ MSD

= ${1 \over {n + 1}}$ $\times$ a units = ${a \over {n + 1}}$ units [$\because$ given, 1 MSD = a units]

Area of rectangle, A = lb = 15.2 $\times$ 2.9 = 44.08 cm²

Minimum possible measurement of scale = 0.1 cm

As we know, the product should have significant figures that are present in the measurement with least number of significant figures which is 2 in this case. So, area measured by scale = 44 cm².

As, RC is the time constant, so it has the dimension of time and V has the dimensions of $L{{dl} \over {dt}}$.

$\therefore$ Dimensions of ${L \over {RCV}},\left[ {{L \over {RCV}}} \right] = {{[L]} \over {[RC][V]}}$

${{[L]} \over {[T][L]\left[ {{A \over T}} \right]}} = {{[L][T]} \over {[T][L][A]}} = {1 \over {[A]}}$

So, $\left( {{L \over {RCV}}} \right)$ has the dimensions $[{A^{ - 1}}]$, i.e. same as the dimensions of ${1 \over {current}}$.