XI-Physics CH-1

Measurements

SQ 1.1

What are base units and derived units? Give some examples of both these units.

Base Units Base units are independent standards from which other units are formed.
Examples metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd).
Derived Units Derived units are obtained by combining base units.
Examples Velocity has unit $\mathrm{m\,s^{-1}}$. Force has unit newton ($\mathrm{N}$), equivalent to $\mathrm{kg\,m\,s^{-2}}$. Energy has unit joule ($\mathrm{J}$), equivalent to $\mathrm{kg\,m^2\,s^{-2}}$.

SQ 1.2

How many significant figures should be retained when multiplying or dividing, and when adding or subtracting numbers?

Multiplication or Division Retain as many significant figures as the factor having the fewest significant figures.
Addition or Subtraction Retain as many decimal places as the term having the fewest decimal places.

SQ 1.3

How is the Vernier scale related to the main scale of a Vernier Callipers? What is meant by L.C. of the Vernier Callipers?

Vernier Scale Relation Usually $n$ Vernier divisions equal $(n-1)$ main-scale divisions.
Least Count The least count is the smallest length that the instrument can reliably measure. Therefore, $$\text{least count}=1\,\text{MSD}-1\,\text{VSD}=\frac{1\,\text{MSD}}{n}.$$

SQ 1.4

Write the following numbers in scientific notation: (a) 143.7 (b) $206.4\times10^7$.

$$143.7=1.437\times10^2,$$

$$206.4\times10^7=2.064\times10^9.$$

In scientific notation, the coefficient must be at least 1 and less than 10: $1\leq a<10$.

SQ 1.5

Write the following numbers using correct prefixes: (a) $580\times10^2\,\mathrm{g}$ (b) $0.45\times10^{-5}\,\mathrm{s}$.

Using the appropriate SI prefixes
Part (a) $$580\times10^2\,\mathrm{g}=58\times10^3\,\mathrm{g}=58\,\mathrm{kg}.$$
Part (b) $$0.45\times10^{-5}\,\mathrm{s}=4.5\times10^{-6}\,\mathrm{s}=4.5\,\mu\mathrm{s}.$$

SQ 1.6

Kinetic energy of a body of mass $m$ moving with speed $v$ is given by $\tfrac12mv^2$. What are the dimensions of kinetic energy?

Mass $$[m]=[M]$$
Velocity $$[v]=[LT^{-1}]$$
Kinetic Energy $$[KE]=[M][LT^{-1}]^2=[ML^2T^{-2}].$$ The required dimensions are mass, length squared, and inverse time squared.

SQ 1.7

How many significant figures are in 37 km, 0.002953 m, 7.50034 cm, and 200.0 m?

37 km 2 significant figures.
0.002953 m 4 significant figures because leading zeros are not significant.
7.50034 cm 6 significant figures.
200.0 m 4 significant figures because the decimal point makes the trailing zeros significant.

SQ 1.8

Write the dimensions of Planck’s constant and angular velocity.

Planck’s Constant From the energy-frequency relation $$E=hf$$ $$[h]=[E][f]^{-1}=[ML^2T^{-1}].$$
Angular Velocity Angle is dimensionless, so angular velocity has the dimensions $$[\omega]=[T^{-1}].$$

CRQ 1.1

Why is it useful to have kilogram and mole as units for the amount of material?

Kilogram It measures the mass of a substance.
Mole It measures the number of entities; $$1\,\mathrm{mol}=6.022\times10^{23}\,\text{entities}$$
Need for Both Equal masses of different substances generally contain different numbers of particles, so both units are required.

CRQ 1.2

A scale has 1 mm minimum divisions. Students record a rod length as 0.4235 m, 0.42 m, and 0.424 m. Which record is correct and why?

Least Count $$1\,\mathrm{mm}=0.001\,\mathrm{m}$$ so the reading should have three decimal places in metres.
Correct Reading $0.424\,\mathrm{m}$ is consistent with the instrument.
Reason $0.4235\,\mathrm{m}$ claims unjustified precision, while $0.42\,\mathrm{m}$ discards available precision.

CRQ 1.3

Why is kilogram, rather than gram, the SI base unit of mass?

SI Base Unit The kilogram was historically selected as the coherent base unit suitable for practical masses.
Coherent Unit Mechanical units follow consistently; for example, $$1\,\mathrm{N}=1\,\mathrm{kg\,m\,s^{-2}}$$ The gram remains a submultiple with a value of $10^{-3}\,\mathrm{kg}$.

CRQ 1.4

For $P=Q+R$, if $Q$ and $R$ both have dimensions $[MLT^{-1}]$, find the dimensions and SI unit of $P$. What if their dimensions differ?

Condition for Addition Only quantities with identical dimensions can be added.
Dimensions of P $$\begin{aligned}{}[P] &= [Q]=[R] \\ &= [MLT^{-1}]\end{aligned}$$
SI Unit $\mathrm{kg\,m\,s^{-1}}$.
Different Dimensions The equation would be dimensionally invalid, so $P$ could not be assigned dimensions from the sum.

CRQ 1.5

What is the least count of a clock with (a) hour, minute, and second hands, and (b) hour and minute hands?

Least Count The smallest marked time interval that can be read directly.
Hour, Minute and Second Hands Normally $1\,\mathrm{s}$.
Hour and Minute Hands Normally $1\,\mathrm{min}$ for a clock with usual graduations.

CRQ 1.6

How can the diameter of a round pencil be measured with a metre rule with accuracy comparable to Vernier callipers?

Method Place $N$ identical pencils tightly side by side and measure their total width $W$.
Diameter of One Pencil $$d=\frac{W}{N}.$$
Advantage Measuring many diameters together reduces fractional uncertainty approximately by the factor $N$.

CRQ 1.7

How would readings differ if a screw gauge were used instead of Vernier callipers to measure a glass plate’s thickness?

Vernier Callipers The least count is typically about $0.1\,\mathrm{mm}$.
Screw Gauge Its smaller least count is typically $0.01\,\mathrm{mm}$, giving more meaningful decimal places and less instrumental uncertainty.
Result After correcting zero error, both measurements should agree within their uncertainties.

CRQ 1.8

How is the correct Vernier reading found when the instrument has a positive zero error of $+0.02\,\mathrm{cm}$?

Observed Reading $$R_{obs}=R_{main}+(n\times LC).$$
Zero Correction A positive zero error requires a negative correction.
Correct Reading $$R_{correct}=R_{obs}-0.02\,\mathrm{cm}.$$
Diagram Values Use the main-scale reading and coinciding Vernier division shown in the exercise figure.

CRQ 1.9

A screw gauge has 50 circular divisions and advances 5 mm in 10 revolutions. Explain how to find the corrected glass-slab thickness when the second division gives a positive zero error.

Pitch $$5/10=0.5\,\mathrm{mm}$$
Least Count $$LC=\frac{0.5}{50}=0.01\,\mathrm{mm}.$$
Zero Error The positive zero error is $$2(0.01)=0.02\,\mathrm{mm}$$
Corrected Thickness For main-scale reading $M$ and circular coincidence $n$ $$t=M+n(0.01)-0.02\,\mathrm{mm}.$$ Apply the negative correction for positive zero error.

CRQ 1.10

What is a dimensionless quantity? Give an example.

Dimensionless Quantity A quantity with no physical dimensions because like dimensions cancel.
Dimensional Formula $[M^0L^0T^0]$.
Examples Refractive index $n$ is the ratio $c/v$. Strain is the ratio $\Delta L/L$. Other examples are coefficient of friction and plane angle measured in radians.

CRQ 1.11

A paper is folded three times and its total thickness is measured with a screw gauge. How is the thickness of one sheet found?

Number of Layers Three folds produce eight layers because the number of layers is $2^3$, which is 8.
Corrected Reading Correct the screw-gauge reading for any zero error.
Thickness of One Sheet $$t_{sheet}=\frac{t_{folded}}{8}.$$ The folded thickness comes from the main and circular scales shown in the figure.

CRQ 1.12

Round 0.02055 and 4656.5 to three significant figures and write them in scientific notation.

For 0.02055 $$\begin{aligned}0.02055 &= 2.055\times10^{-2} \\ 0.02055\text{ (to 3 s.f.)} &= 2.06\times10^{-2}\end{aligned}$$
For 4656.5 $$\begin{aligned}4656.5 &= 4.6565\times10^3 \\ 4656.5\text{ (to 3 s.f.)} &= 4.66\times10^3\end{aligned}$$
Rule Use the first discarded digit to round the last retained digit.