2.

UNITS AND MEASUREMENT

1. MEASUREMENT, SYSTEMS OF UNITS & FUNDAMENTAL UNITS

Main points from the topic:

Physical quantity: A quantity which can be measured and expressed in the

form of laws is called a physical quantity.
Physical quantity (Q) = Magnitude (n) x Unit (u)
- In this part, n is the numerical value and u is the unit. As the unit changes,
the magnitude will also change but the product of these two will remain
the same.
- For example, 2 cm can be written as 0.02 m, in this case, these two
quantities are equal to each other.

Fundamental and Derived units

Any unit of mass, length and time in mechanics is called a fundamental,
absolute or base unit. Other units which can be expressed in terms of
fundamental units are called derived units.
System of units: A complete set of fundamental and derived units for all
kinds of physical quantities is called the system of units.
There are four systems of units:
i. CGS system (centimetre, gram and second)
ii. MKS system (metre, kilogram and second)
iii. FPS system (feet, pound and second)
iv. S.I system (rationalised MKS system)
S.I system: It is known as the international system of units. There are seven
fundamental quantities in this system.
a. Metre: The metre is the length of the path travelled by light in a
vacuum during a time of 1/299,792,458 of a second.
b. Kilogram: The kilogram is equal to the mass of the international
prototype of the kilogram platinum-iridium alloy cylinder) kept at the
International Bureau of Weights and Measures, at Se near Paris,
France.
c. Second: The second is the duration of 9,192,631,770 periods of the
radiation corresponding to the transition between the two hyperfine
levels of the ground state of the caesium-133 atom.
d. Ampere: The ampere is that constant current which, if maintained in
two straight parallel conductors of infinite length, of negligible circular
cross-section, and placed 1 metre apart in vacuum, would produce
between these conductors a force equal to 2 x 10-7 Newton per metre of
length.
e. Kelvin: The kelvin, is the fraction 1/273.16 of the thermodynamic
temperature of the triple point of water.
 f. Mole: The mole is the amount of substance of a system, which
contains as many elementary entities as there are atoms in 0.012
kilograms of carbon-12.
g. Candela: The candela is the luminous intensity, in a given direction,
of a source that emits monochromatic radiation of frequency 54010 12
hertz and that has a radiant intensity in that direction of 1/683 watt per
steradian.

Besides the above seven fundamental units, two supplementary units

are also defined - Radian (rad) for plane angle and Steradian (sr) for
solid angle.

Significant figures
- Every measurement results in a number that includes reliable digits and
uncertain digits. Reliable digits plus the first uncertain digit are
called significant digits or significant figures. These indicate the
precision of measurement which depends on the least count of measuring
instrument.
- example, the period of oscillation of a pendulum is 1.62 sec. Here 1 and 6
are reliable and 2 is uncertain. Thus, the measured value has three
significant figures.
- Rules for determining the number of significant figures
a. All non-zero digits are significant.
b. All zeros between two non-zero digits are significant irrespective of
the decimal place.
c. For a value less than 1, zeroes after the decimal and before non-zero
digits are not significant. Zero before decimal place in such a number
is always insignificant.
d. Trailing zeroes in a number without a decimal place are insignificant.
e. Trailing zeroes in a number with decimal place are significant.
- Cautions to remove ambiguities in determining the number of
significant figures
a. Change of units should not change the number of significant digits.
Example, 4.700m = 470.0 cm = 4700 mm. In this, the first two
quantities have 4 but the third quantity has 2 significant figures.
b. Use scientific notation to report measurements. Numbers should be
expressed in powers of 10 like a x 10b where b is called the order of
magnitude. Example, 4.700 m = 4.700 x 102 cm = 4.700 x 103 mm =
4.700 x 10-3 In all the above, since power of 10 are irrelevant, number
of significant figures are 4.
c. Multiplying or dividing exact numbers can have an infinite number of
significant digits. Example, radius = diameter / 2. Here 2 can be
written as 2, 2.0, 2.00, 2.000 and so on.
Explain what you learned from the above topic. Then try answering the following
questions.
1. Physical quantities are
a. quantities such as degrees, radians and steradians
b. quantities such as length, mass, time, electric current, thermodynamic
temperature, amount of substance, and luminous intensity
c. quantities such as pounds, dollars and rupees
d. quantities such as kilos, pounds and gallons
Answer: Option b

Explanation: Physical quantities are quantities such as length, mass, time,

electric current, thermodynamic temperature, amount of substance, and
luminous intensity.

2. Electron volt is a unit of

a. Charge
b. potential difference
c. energy
d. magnetic force
Answer: option c

Explanation: It is the unit of energy.

3. The volume of a cube in m³ is equal to the surface area of the cube in m². The
volume of the cube is
a. 64 m³
b. 216 m³
c. 512 m³
d. 196 m³
Answer: Option

4. In the SI system the fundamental units are

a. meter, kilogram, second, ampere, Kelvin, mole and candela
b. meter, kilogram, second, coulomb, Kelvin, mole and candela
c. meter, Newton, second, ampere, Kelvin, mole and candela
d. meter, kilogram, second, ampere, Kelvin, mole
Answer: Option a

Explanation: In the SI system the fundamental units are meter, kilogram,

second, ampere, Kelvin, mole and candela

5. Light year is a unit of

a. Time
b. Distance
c. sunlight intensity
d. mass
 Answer: Option b

Explanation: Light year is the unit of distance which is used to measure

astronomical distances. It is represented by ly.
1 ly = 9.461 x 1012 m

6. The weight of a body is 12g. This statement is not correct because

a. The correct symbol for the unit of weight has not been used.
b. The correct symbol for gram is gm.
c. The weight should be expressed in kg.
d. For some reason other than those given above.
Answer: Option a

Explanation: The unit of mass is gram (g) and the unit of weight is newton (N).

7. If the unit of force and length is doubled, the unit of energy will be
a. 1/2 times
b. 2 times
c. 4 times
d. 1/4 times
Answer: Option c

Explanation: The formula of energy can be defined as work done which has an
equivalent,
Energy = force x distance
E = 2N x 2m = 4 N-m = 4 joules

8. Which of the following numerical values has three significant figures?

a. 5.055
b. 0.050
c. 50.50
d. 0.500
Answer: Option d

Explanation: The digit 0.500 has three significant figures, 5.055 and 50.50 have
four significant figures, and 0.0050 has two significant figures.

9. What is the number of significant figures in (3.20 + 4.80) × 105?

a. 2
b. 3
c. 4
d. 5
Answer: Option b

Explanation: (3.20 + 4.80) x 105 = 8.00 x 105 has three significant figures.
10. Which of the following is not the name of a physical quantity?
a. Kilogram
b. Density
c. Energy
d. Impulse
Answer: Option a

Explanation: Kilogram is not a physical quantity but the unit of physical

quantity which is called mass.
2. DIMENSIONAL ANALYSIS

Main points from the topic:

Dimensions of a physical quantity

When a derived quantity is expressed in terms of fundamental quantities, it is
written as a product of different powers of the fundamental quantities.
The powers to which fundamental quantities must be raised to express the given
physical quantity are called its dimensions.
Important dimensions of complete physics
a. Mechanics
b. Heat
c. Quantities having the same dimensions

Application of Dimensional analysis:

i. To find the unit of a physical quantity in a given system of units.
ii. To find dimensions of physical constant or coefficients.
iii. To convert a physical quantity from one system to the other.
iv. To check the dimensional correctness of a given physical relation: This
is based on the principle of homogeneity. According to this principle,
the dimensions of each term on both of an equation must be the same.
v. To derive new relations.

Limitations of Dimensional Analysis:

i. If dimensions are given, the physical quantity may not be unique.
ii. Numerical constant having no dimensions cannot be deduced by the
methods of dimensions.
iii. The method of dimensions can not be used to derive relations other than
the product of power functions.
For example,
 𝑎𝑡 2
𝑠 = 𝑢𝑡 + 𝑜𝑟 𝑦 = 𝑎 𝑠𝑖𝑛𝜔𝑡
2
iv. The method of dimensions cannot be applied to derive a formula
consisting of more than three physical quantities.
Explain what you learned from the above topic. Then try answering the following
questions.
1. Which of the following pairs has the same dimensions?
a. specific heat and latent heat
b. impulse and momentum
c. surface tension and force
d. moment of inertia and torque
Answer: Option b

Explanation: Impulse has units equal to kg-m/s and momentum has the same
units kg-m/s.

2. The dimensions of kinetic energy are

a. [M2 L2 T]
b. [ML2 T]
c. [ML2 T-2]
d. [ML2 T-1]
Answer: Option c

Explanation: The SI unit of kinetic energy is kg-m2/s2 whose dimensions will

become [ML2 T-2]

3. A force F is given by F = at + bt², where t is time. What are the dimensions of a and
b?
a. MLT-1 and MLT0
b. MLT-3 and ML2T4
c. MLT-4 and MLT
d. MLT-3 and MLT-4
Answer: Option d

Explanation: The rule of the dimension analysis is that the same dimensional
quantities can be written equal to each other and added all together. Here, the
dimension of force is [MLT-2], hence, at will have the same dimension which
makes,
at = [MLT-2]
a = [MLT-2]/[T] = [ML T-3]

bt2 = [MLT-2]
b = [MLT-2]/[T2]

4. The atmospheric pressure is 106 dyne/cm². What is its value in the SI unit?
a. 105 newton/m²
b. 106 newton/m²
c. 104 newton/m²
d. 103 newton/m²
Answer: Option a
 Explanation: The atmospheric pressure is 106 dyne/cm², in N/m2 it is,
1 dyne = 10-5 N and 1 cm2 = 10-4 m2
106 dyne/cm² = 106 x 10-5 N / 10-4 m² = 105 newton/m²

5. In a system of units, if force (F), acceleration (A) and time (T) are taken as
fundamentals units then the dimensional formula of energy is
a. FA2 T
b. FAT2
c. F2AT
d. FAT
Answer: Option b

Explanation: The energy can be written as

Energy = force x distance
[energy] = FAT2

6. If force (F), work (W) and velocity (v) are taken as fundamental quantities. What is
the dimensional formula of time (T)?
a. [WFv]
b. [WFv-1]
c. [W-1 F-1 v]
d. [WF-1 v-1]
Answer: Option d

Explanation: In the dimensions, [WF-1 v-1], everything will be cancelled out and
the final form will remain the seconds (sec.) only.

7. The dimensions of kinetic energy are the same as that of

a. Force
b. Pressure
c. Work
d. Momentum
Answer: Option c

Explanation: Both work and energy have the same SI units, hence, the
dimensions are the same.

8. Which of the following groups have different dimensions?

a. Potential difference, EMF, voltage
b. Pressure, stress, Young’s modulus
c. Heat, energy, work done
d. Dipole moment, electric flux, electric field
Answer: Option d

Explanation: The dipole moment, electric flux and electric field have different
dimensions. About these quantities, you will learn in the XII class.
9. [ML-1 T-2] is the dimensional formula of
a. magnetic induction
b. self-inductance
c. electric potential
d. electric field
Answer: Option a

Explanation: The unit of magnetic induction is Tesla (T) whose dimension is

[ML-1 T-2].

10. What is the dimensional formula of the magnetic field?

a. MT-2 A-1
b. MT-1 A-2
c. M-1 L-2 TA-1
d. M-1 LTA-2
Answer: Option a

Explanation: The SI unit of the magnetic field is N/m/A which can be solved
further and find the dimensions MT-2 A-1