Units and measurements are essential for expressing physical quantities in a clear and standardized form, forming the backbone of all scientific calculations. This chapter focuses on building strong conceptual understanding through carefully designed questions and answers, along with practice exercises. By mastering these concepts, students can improve their accuracy, understand measurement uncertainties, and confidently solve physics problems.
Conceptual Questions and Answers Based On Units and Measurements
Q. What is meant by unit?
Ans. The unit of a physical quantity is a standard of the same kind chosen in order to measure that quantity.
Q. What should we know in order to measure a physical quantity?
Ans. Its unit and the number of times the unit is contained in the physical quantity.
Q. Is the measure of a physical quantity dependent upon the choice of unit?
Ans. If the size of the unit chosen to measure the physical quantity is bigger, the numerical value of the physical quantity is smaller and vice-versa. However, the measure of a physical quantity remains the same.
Q. How many nanometre are there in one metre?
Ans. Now,
$1$ nanometre (nm) $= 10^{-9}$ m
$$1\ \text{m} = 10^{9}\ \text{nm}$$
Q. What is the unit for measuring wavelength of light?
Ans. Angstrom (Å).
$1\ \text{Å} = 10^{-10}\ \text{m}$.
Q. How many angstrom are there in one metre?
Ans. Now,
$1$ angstrom (Å) $= 10^{-10}$ m
$$\therefore\quad 1\ \text{m} = 10^{10}\ \text{Å}$$
Q. How many fermi are there in one metre?
Ans. Now,
$1$ fermi (f) $= 10^{-15}$ m
$$\therefore\quad 1\ \text{m} = 10^{15}\ \text{f}$$
Q. Is light year a unit of time?
Ans. No, it is a unit of distance.
Q. Define light year.
Ans. One light year is defined as the distance travelled by light in one year.
$$1\ \text{light year} = 9.46 \times 10^{15}\ \text{m}$$
Q. How many light years are there in one metre?
Ans. Now,
$$\begin{aligned}
1\ \text{light year} &= 3 \times 10^{8}\ \text{m/s} \times 1 \times (365 \times 24 \times 60 \times 60)\ \text{s} \
&= 9.46 \times 10^{15}\ \text{m}
\end{aligned}$$
$$1\ \text{m} = \frac{1}{9.46 \times 10^{15}} = 1.057 \times 10^{-16}\ \text{light year}$$
Q. Define astronomical unit.
Ans. It is the mean distance of the sun from the earth.
$$1\ \text{AU} = 1.496 \times 10^{11}\ \text{m}$$
Q. Give approximate ratio of 1 AU and 1 light year.
Ans. Now, $1\ \text{AU} = 1.496 \times 10^{11}$ m and $1\ \text{light year} = 9.46 \times 10^{15}$ m
$$\therefore\quad \frac{1\ \text{AU}}{1\ \text{light year}} = \frac{1.496 \times 10^{11}}{9.46 \times 10^{15}} \approx 10^{-5}$$
Q. Define a parsec.
Ans. It is the distance at which an arc of length one astronomical unit subtends an angle of one second of an arc.
$$1\ \text{parsec} = 3.08 \times 10^{16}\ \text{m}$$
Q. Express one parsec in terms of light years.
Ans. Now, $1\ \text{parsec} = 3.08 \times 10^{16}$ m and $1\ \text{light year} = 9.46 \times 10^{15}$ m
Therefore, number of light years in 1 parsec
$$\frac{3.08 \times 10^{16}}{9.46 \times 10^{15}} = 3.256$$
Q. Arrange the following units of length in descending order: light year; astronomical unit and parsec.
Ans. parsec; light year and astronomical unit.
Q. How many times a kg is larger than a mg ?
Ans.
$$\frac{1\ \text{kg}}{1\ \text{mg}} = \frac{10^{3}\ \text{g}}{10^{-3}\ \text{g}} = 10^{6}$$
Q. How many quintals are there in a gigagram ?
Ans.
$$\frac{1\ \text{giga gram}}{1\ \text{quintal}} = \frac{10^{9}\ \text{g}}{10^{2}\ \text{kg}} = \frac{10^{9}\ \text{g}}{10^{2} \times 10^{3}\ \text{g}} = 10^{4}$$
Q. The mass of an electron is $9.11 \times 10^{-31}$ kg. How many electrons would make 1 kg?
Ans. Number of electrons in 1 kg $= \dfrac{1\ \text{kg}}{\text{mass of electron}}$
$$= \frac{1}{9.11 \times 10^{-31}} = 1.1 \times 10^{30}$$
Q. The mass of a proton is $1.67 \times 10^{-27}$ kg. How many protons would make 1 g?
Ans. Number of protons in 1 g $= \dfrac{1\ \text{g}}{\text{mass of proton}}$
$$= \frac{10^{-3}\ \text{kg}}{1.67 \times 10^{-27}\ \text{kg}} = 5.99 \times 10^{23}$$
Q. Define atomic mass unit (a.m.u.).
Ans. One a.m.u. is defined as $\dfrac{1}{12}$ th of the mass of one $^{12}$C atom.
$$1\ \text{a.m.u.} = \frac{1}{12} \left( \frac{12 \times 10^{-3}\ \text{kg}}{6.02 \times 10^{23}} \right) = 1.66 \times 10^{-27}\ \text{kg}$$
Q. How many kilograms are there in 1 a.m.u ?
Ans. $1$ a.m.u. $= 1.66 \times 10^{-27}$ kg
Q. How much larger than microsecond is a millisecond?
Ans. $1$ millisecond $= 10^{-3}$ s
$1$ microsecond $= 10^{-6}$ s
$$1\ \text{millisecond} = 10^{3}\ \text{microsecond}$$
Q. How much larger than a nanosecond is a millisecond?
Ans. $1$ millisecond $= 10^{-3}$ s
$1$ nanosecond $= 10^{-9}$ s
$$1\ \text{millisecond} = 10^{6}\ \text{nanosecond}$$
Q. What is meant by mean solar second ?
Ans. In astronomy and timekeeping, a mean solar second is the standard unit of time derived from the average length of a solar day. A solar day is the time it takes for the Earth to rotate once on its axis so that the Sun appears at the same meridian (the highest point in the sky). However, because Earth’s orbit is elliptical and its axis is tilted, the length of an actual solar day varies throughout the year. To create a consistent timekeeping system, scientists use the Mean Solar Day, which is the average of all solar days over a full year. The mean solar second is defined as a specific fraction of this average day :
1 Mean Solar Day = 24 hours
24 hours = 1,440 minutes
1,440 minutes = 86,400 seconds
Therefore, a mean solar second is exactly
$$\frac{1}{86,400}$$
of a mean solar day.
Q. How many nanoseconds are there in 1 shake ?
Ans. Now, $1$ shake $= 10^{-8}$ s and $1$ ns $= 10^{-9}$ s
$$\frac{1\ \text{shake}}{1\ \text{ns}} = \frac{10^{-8}\ \text{s}}{10^{-9}\ \text{s}} = 10$$
Q. Express 0.000003 m as a power of 10.
Ans. $0.000003 = 3 \times 10^{-6}$
Q. Express 0.000003 m as a power of 10.
Ans. $0.000003 = 3 \times 10^{-6}$
Q. What necessitated the selection of some fundamental units ?
Ans. For measuring mass, length and time, three independent units i.e. kilogram, metre and second are used. For measuring other physical quantities, if a separate unit is defined for each of them, then it will become very difficult to remember all of them as they will be quite unrelated to each other.
Q. 2.02. What are the characteristics of a physical standard?
Ans.
- It should be well defined.
- It should be of suitable size i.e. neither too large nor too small in comparison to the quantity to be measured.
- It should be easily reproducible at all places.
- It should not change with time and from place to place.
- It should not change with change in its physical conditions, such as temperature, pressure, etc.
- It should be easily accessible.
Q. The average wavelength of the yellow light emitted from a sodium lamp is 5,893 Å. Express it in nm.
Ans. Here, $\lambda = 5,893$ Å $= 5,893 \times 10^{-10}$ m
Since $1$ nm $= 10^{-9}$ m, it follows that
$$\lambda = \frac{5,893 \times 10^{-10}}{10^{-9}} = 589.3\ \text{nm}$$
Q. Express the distance from earth to sun in (i) parsec and (ii) light year.
Ans. The distance of the earth from the sun $= 1$ AU $= 1.496 \times 10^{11}$ m
(i) Distance of the earth from the sun (in parsec)
$$= \frac{1.496 \times 10^{11}}{3.08 \times 10^{16}} = 4.857 \times 10^{-6}\ \text{parsec}$$
(ii) Distance of the earth from the sun (in light year)
$$= \frac{1.496 \times 10^{11}}{9.46 \times 10^{15}} = 1.581 \times 10^{-5}\ \text{light year}$$
Q. What is the difference between Å and AU ?
Or
Do Å and AU stand for the same unit of length ?
Ans. No, Å and AU do not stand for the same unit of length.
$1$ Å (angstrom) $= 10^{-10}$ m
and $1$ AU (astronomical unit) $= 1.496 \times 10^{11}$ m
Q. Are there more nanoseconds in a second than the number of seconds in a year?
Ans. Number of nanoseconds in a second
$$= \frac{1\ \text{s}}{10^{-9}\ \text{s}} = 10^{9}$$
Number of seconds in a year $= 365 \times 24 \times 60 \times 60 = 3.1536 \times 10^{7}$
Therefore, number of nanoseconds in a second is more than the number of seconds in a year.
Q. Is the measure of an angle dependent upon the unit of length?
Ans. The angle (in radian) subtended by an arc of a circle at its centre is defined as the ratio of the length of the arc to its radius i.e.
$$\theta\ (\text{in radian}) = \frac{\text{length of arc}}{\text{radius}} = \text{no dimension}$$
Hence, the measure of an angle is independent of the unit of length used to measure the angle.
Q. What is a coherent system of units ?
Ans. A coherent system of units is a system where the units for derived physical quantities are obtained exclusively by the multiplication or division of fundamental (base) units, without the need for any numerical conversion factors other than 1. In such a system, the equations between units look exactly like the equations between the physical quantities themselves.
Key Characteristics
No Arbitrary Constants: You don’t need to multiply by extra numbers (like 60, 12, or 3.14) to define a new unit.
Simple Relationships: If you have base units for length ($L$) and time ($T$), the coherent unit for velocity is simply $L/T$.
The SI Example: The International System of Units (SI) is the most famous coherent system.
Examples of Coherence
1. Force (The Newton)
In the SI system, the formula for force is $F = ma$.
Base unit of Mass: kg
Base unit of Acceleration: m/s²
Coherent Unit: $1\text{ Newton (N)} = 1\text{ kg} \times 1\text{ m/s}^2$Because the constant is exactly 1, the Newton is a coherent unit.
2. Power (The Watt)
Power = Work / Time
Coherent Unit: $1\text{ Watt (W)} = 1\text{ Joule (J)} / 1\text{ second (s)}$Again, since no numerical factor is needed, the Watt is coherent within the SI system.
Non-Coherent Example
To understand coherence, it helps to see what it isn’t.
Area in Acres: If your base unit of length is the foot, the coherent unit of area is the square foot.
An Acre is not coherent because $1\text{ Acre} = 43,560\text{ square feet}$. You have to multiply by a complex number (43,560) to get the unit, making it non-coherent.
Q. Derive the SI unit of joule (J) in terms of fundamental units. (P.S.S.C.E. 1991)
Ans. Joule is unit of work. We know that
$$\text{work} = \text{force} \times \text{distance} = \text{mass} \times \text{acceleration} \times \text{distance} = \frac{\text{mass} \times \text{distance}^{2}}{\text{time}^{2}}$$
$$\therefore\quad J = \frac{\text{kg} \times \text{m}^{2}}{\text{s}^{2}} = \text{kg}\ \text{m}^{2}\ \text{s}^{-2}$$
Q. What is the advantage in choosing the wavelength of a light radiation as a standard of length ?
Ans. The advantages in choosing the wavelength of a light radiation as a standard of length are as follows:
- The wavelength of light is not affected by time and environment.
- This standard of length does not undergo any change with place.
Q. Both mks and SI systems have metre, kilogram and second as the basic units. How does mks system differ from SI system ?
Ans. SI system covers the units of physical quantities from all the branches of physics, whereas mks system is confined to mechanics only.
Q. In defining the standard of length, we have to specify the temperature at which the measurement should be made. Are we justified in calling length a fundamental quantity, if another physical quantity (temperature) has to be specified in choosing a standard?
Ans. The length of an object varies with the temperature but the standard for length, which is now defined in terms of wavelengths of light, is not affected by temperature. Hence, we are justified in calling length a fundamental quantity.
Q. How would you define a unit of mass, if people had placed a standard spring at Sevres instead of a standard mass?
Ans. When a mass is suspended from a spring, the frequency of the oscillating mass varies inversely as the square root of the oscillating mass. To eliminate the effect of air friction on the oscillating motion of the mass, it may be suspended in vacuum. It may be pointed out that this method will give the measure of inertial mass.
Q. How many wavelengths of Kr$^{86}$ are there in one metre?
Ans. The light emitted by krypton-86 has wavelength $6057\ 021$ Å or $6.057\ 021 \times 10^{-7}$ m.
Therefore, no. of wavelengths of Kr$^{86}$ in one metre
$$1\ \text{m} = \frac{1}{6.057\ 021 \times 10^{-7}} = 1\ 650\ 763.3$$
Q. What is a physical quantity?
Ans. A physical quantity is any measurable property that can be expressed using a numerical value and a unit, such as length, mass, and time.
Q. What are fundamental and derived quantities?
Ans. Fundamental quantities are basic and independent (e.g., length, mass, time). Derived quantities are obtained from fundamental quantities using mathematical relationships (e.g., velocity, force).
Q. What is the SI system of units?
Ans. The SI system is an internationally accepted system of units that standardizes measurements. It includes seven base units such as meter (m), kilogram (kg), and second (s).
Q. Why are standard units necessary?
Ans. Standard units ensure uniformity and consistency in measurements across different regions and experiments, making scientific communication reliable.
Q. What is dimensional analysis?
Ans. Dimensional analysis is a method to check the correctness of equations and to derive relationships between physical quantities by comparing their dimensions.
Q. What is the difference between accuracy and precision?
Ans. Accuracy refers to how close a measurement is to the true value, while precision indicates the consistency of repeated measurements.
Q. What are significant figures?
Ans. Significant figures are the digits in a measurement that carry meaningful information about its precision, including all certain digits and the first uncertain digit.
Q. What is absolute error?
Ans. Absolute error is the difference between the measured value and the true value of a quantity:
$$
\text{Absolute Error} = | \text{Measured Value} – \text{True Value} |
$$
Q. What is relative error?
Ans. Relative error is the ratio of absolute error to the true value:
$$
\text{Relative Error} = \frac{\text{Absolute Error}}{\text{True Value}}
$$
Q. What is percentage error?
Ans. Percentage error is the relative error expressed as a percentage:
$$
\text{Percentage Error} = \text{Relative Error} \times 100%
$$
Q. What is least count of an instrument?
Ans. The least count is the smallest value that can be measured accurately using a measuring instrument.
Q. What are systematic and random errors?
Ans. Systematic errors occur due to flaws in instruments or methods and affect all measurements consistently. Random errors arise from unpredictable variations and differ from one measurement to another.
Q. A length is measured as 10.2 cm, while the true value is 10.0 cm. Calculate the absolute and percentage error.
Ans. Absolute Error:
$$
|10.2 – 10.0| = 0.2 \text{ cm}
$$
Relative Error:
$$
\frac{0.2}{10.0} = 0.02
$$
Percentage Error:
$$
0.02 \times 100 = 2%
$$
Practice Exercise
- What is a unit? Explain the process of measuring a physical quantity.
- What do you understand by the unit of measurement? Show that bigger is the unit, smaller is the numerical value of physical quantity and vice versa.
- What is a standard? Name the coherent system of units for measuring various physical quantities for mechanics only. Hence, define metre, kilogram and second.
- What are fundamental units? State how metre has been defined in terms of wavelength and second in terms of periods of radiation.
- What is the SI unit of length? Why this unit is defined in terms of wavelength of light radiation?
- Name the basic and supplementary units of SI. How are the three units of mass, length and time defined on SI?
- What are the main characteristics of SI?
- Briefly comment on the advantage of atomic standards of length and time over the conventional standards.
- Name basic units of a system used for measuring all physical quantities. What is meant by tesla, nano and femto?
- Define the following: (i) Light year (ii) Parsec and (iii) Astronomical unit.
- Define astronomical unit, light year and parsec. How are these related to each other?
- What is meant by the unit of a physical quantity? Define the basic units of SI. Why the unit of length has been defined in terms of the wavelength and the unit of time in terms of the periods of a radiation?
- (a) What are the characteristics of a standard unit?
(b) Define the units of length: parsec, light year and astronomical unit.
(c) Define atomic mass unit and mean solar second. - What are the basic and supplementary units of SI? Define each of them. Also give the advantages of SI over the other systems of units.
- Name the various systems of units and discuss them briefly. Define the basic and supplementary units of SI. What are the merits of SI over other systems of units?
Important Chapter Links
In this chapter on Units and Measurements: Conceptual Questions and Answers, Practice Exercise, you will develop a solid foundation in measurement principles, SI units, and error analysis. The section includes important conceptual questions with clear explanations, followed by practice exercises to reinforce learning. It is designed to help students improve precision in calculations and build confidence for board and JEE exams and problem-solving.