Units, Dimensions and Measurements MCQ Quiz - Objective Question with Answer for Units, Dimensions and Measurements - Download Free PDF

Calculation:

\(\mathrm{L}=\frac{\mathrm{pn}^{4}}{\mathrm{~pq}}\)

\(\Rightarrow \frac{\Delta \mathrm{L}}{\mathrm{L}} \times 100=\left[\frac{\Delta \mathrm{p}}{\mathrm{p}}+4 \frac{\Delta \mathrm{n}}{\mathrm{~n}}+\frac{\Delta \mathrm{p}}{\mathrm{p}}+\frac{\Delta \mathrm{q}}{\mathrm{~q}}\right] \times 100 \)

\(\Rightarrow \frac{\mathrm{11x}}{10}=\left[\frac{3}{60}+4\left(\frac{0.1}{20}\right)+\left(\frac{0.2}{40}\right)+\frac{0.1}{50}\right] \times 100\)

⇒ x = 7

Concept:

Vernier Caliper

• Vernier Caliper is a precision instrument that can be used to measure internal and external distances accurately.
• A vernier caliper consists of two main parts:
• the main scale engraved on a solid L-shaped frame and the vernier scale that can slide along the main scale. 
• It works on the principle of vernier and can measure the dimensions to an accuracy of 0.02 mm.

Parts of a Vernier caliper:

• Outside jaws: Used to measure the external diameter or width of an object
• Inside jaws: Used to measure the internal diameter of an object
• Locking screw: to lock the jaws
• Adjusting screw: to take an accurate measurement of the workpiece.

Explanation:

In general one vernier scale division is smaller than one main scale division but in some modified cases it may be not correct.

Also least count is given by one main scale division / number of vernier scale division for normal vernier calliper.

Calculations:

Given:

Main scale readings: 5.10 cm and 5.15 cm

Vernier scale reading: 24 divisions

1. Main Scale Division (MSD):

The main scale can measure from 5.10 cm to 5.15 cm, indicating a total length of 0.05 cm (5.15 cm - 5.10 cm).

Since there are 50 divisions on the main scale, each main scale division (MSD) is: 0.05 cm / 50 = 0.001 cm.

2. Vernier Scale Division (VSD):

Each vernier scale division (VSD) corresponds to: 1 MSD - (1 VSD) = 0.001 cm.

Therefore, VSD = 0.001 cm.

3. Least Count (LC):

The least count is the difference between one main scale division and one vernier scale division: LC = MSD - VSD = 0.001 cm - 0.001 cm = 0.000 cm.

However, considering the measurement precision, the least count is effectively: LC = 0.001 cm.

4. Diameter Measurement:

Using the least count, the measured diameter is: 5.10 cm + (24 divisions × 0.001 cm/division) = 5.10 cm + 0.024 cm = 5.124 cm.

The measured diameter is approximately 5.124 cm.

Concept Used:

The error in measuring the emf can be calculated using the formula:

V = iR = E / (r + R)

Calculation:

Given:

Resistance of voltmeter, R = 998 ohms

Internal resistance of the cell, r = 2 ohms

Emf of the cell, E = 2 volts

The error in emf measurement is:

V = iR = E / (r + R)

V = (2 / (2 + 998)) × 998 = 1.996 V

ΔV = 2 - 1.996 = 0.004 = 4 × 10^-3 V

Hence, option C is the correct answer.

Calculation:

The given physical quantity \(A = \dfrac{a^2b^3}{c\sqrt{d}}\)

Given that  \(\dfrac{\Delta a}{a}\times 100 = 1\%\), \(\dfrac{\Delta b}{b}\times 100 = 3\%\), \(\dfrac{\Delta c}{c}\times 100 = 2\%\) and \(\dfrac{\Delta d}{d}\times 100 = 2\%\)

Percentage error in \(A\),

\(\dfrac{\Delta A}{A}\times 100 = 2\dfrac{\Delta a}{a}\times 100 + 3\dfrac{\Delta b}{b}\times 100 + \dfrac{\Delta c}{c}\times 100 + \dfrac{1}{2}\times \dfrac{\Delta d}{d}\times 100\)

\(\implies \dfrac{\Delta A}{A}\times 100 = 2\times 1 + 3\times 3 + 2 + \dfrac{1}{2}\times 2 = 14\%\)

The correct answer is option 1) i.e. joule (J)

CONCEPT:

• Work function: It is the minimum amount of energy required to cause photo-emission of electrons from a metal surface when light is incident on it.
  • The work function is also known as the threshold energy.
  • The energy of the incident light is equal to the sum of the work function and the photoelectron kinetic energy.

  • Therefore, the total energy of photon = work function + maximum kinetic energy of the electron.

The energy of a photon is given by the equation:

 \(E =hν\)

Where ν is the frequency of incident light and h is the Planck's constant.​

EXPLANATION:

• The work function is a measure of energy. All forms of energy have the same SI unit i.e. joule (J).

CONCEPT:

• Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit, and a Dimension is a mathematical tool used for studying the nature of physical quantities. 
• The basic concept of dimensions is that we can add or subtract only those quantities which have the same dimensions.
• And the dimensional formula is defined as the expression of the physical quantity in terms of mass, length, and time.

​EXPLANATION:

Strain:

•    The ratio of change in configuration to the original configuration is called strain.

• As the strain is the ratio of two like quantities, it has no units and no dimensions.
•  and has no dimension.
• Its dimension can be expressed as M0L0T0.

​The correct answer is M0L0T0​.   

CONCEPT:

• Principle of homogeneity of dimensions: According to this principle, a physical equation will be dimensionally correct if the dimensions of all the terms occurring on both sides of the equation are the same.
  • This principle is based on the fact that only the physical quantities of the same kind can be added, subtracted, or compared.
  • Thus, the velocity can be added to velocity but not to force.

EXPLANATION:

Given that:

F = ax2 + bx + c

• From the principle of dimensional homogeneity, the left-hand side of the equation dimensionally equal to the right-hand side of the equation.

The dimensional formula of Force (F) = [M L T^-2]

The dimensional formula of Displacement (x) = [L]

LHS = RHS

[M L T-2] = [b] ×  [L]

[b] = M L⁰ T^-2

Hence option 4 is correct.

The correct answer is 27 K.

Concept:

Celsius scale

• The Celsius temperature scale is also known as the Centigrade temperature scale because it divides the boiling and freezing points of water by 100 degrees.
• The Celsius temperature scale is used to measure temperatures all over the world.
• The temperatures on this scale are all in °C (degree celsius).

Kelvin scale

• The Kelvin temperature scale is also known as the thermodynamic scale.
• The Kelvin scale was designed in such a way that the zero point of the temperature scale is set at absolute zero. 
• As a result, the absolute zero is 0 K. 
• The Kelvin temperature scale is widely used in scientific calculations and equations because it has a direct relationship with absolute zero.
• The temperatures on this scale are in K (Kelvin).

Calculation:

The relation between Kelvin and Celsius scale can be written as:

T (K)  = T (°C)  + 273
Let T₁ = a₁ °C = (a₁ + 273) K
      T₂ = a₂ °C = (a₂ + 273) K
Change in temperature:
⇒ T₂ - T₁ = (a₂ - a₁)°C = (a₂ - a₁) K
Temperature change in Celsius scale = Temperature change in Kelvin scale = 27 K

CONCEPT:

• The standard units of measurement defined by the ISU for the seven base quantities are SI base units.
• All other SI units are derived from them.
• 7 Basic SI units with their quantities: 

EXPLANATION:

• From the above table, it is clear that Candela is a unit of luminous intensity. Therefore option 4 is correct. 

Important Points

• Supplementary units: The units that are used along with base units to form derived units in the International System are called supplementary units.

CONCEPT:

• A physical quantity is a property of a material. It can be expressed in number by measurement.
• A physical quantity is expressed by a numerical value and a unit. For example, the physical quantity length can be expressed as 4 meters, where 4 is the numerical value and meter is the unit.
• The SI units are the standard units of measurement defined by the International System of Units (SI).

EXPLANATION:

• Following are the SI units of the physical quantity:
   

The kilogram represents the unit of a physical quantity and not the physical quantity. Therefore option 4 is correct.

• The name and symbol of SI units are written in lowercase.
• Except for the symbols of those SI units, named after a person, which are written with an initial capital letter.
• For example, the second has the symbol s, but the kelvin has the symbol K because it is named after Lord Kelvin.

Concept:

• Density: The mass per unit volume is called density.

Density (ρ) = Mass (m)/Volume (m3)

• The SI unit of density is kg/m3.

Calculation:

Given: Mass = 13 g = 13 × 10^-3 Kg

Volume = 1 cm³ = (1 × 10^-2)³ = 1 × 10^-6 m³

In SI unit

\(Density = \frac{{13\;g}}{{c{m^3}}} = \frac{{13 × {{10}^{ - 3}}kg}}{{{{10}^{ - 2}}^3{m^3}}} = \frac{{13 × {{10}^{ - 3}}kg}}{{{{10}^{ - 6}}{m^3}}} = 13 × 10^{3}\;kg/{m^3}\)

Concept-

The dimensional formula is defined as the expression of the physical quantity in terms of mass, length, time and ampere.

Explanation-

Power – It is defined as rate of doing work.

 \(\therefore P = \frac{W}{t}\)

Where, P = power, W = work done and t = time.

Now,

Dimensional formula of work (W) = [ML²T^-2]

Dimensional formula of time (t) = [T¹]

\(P = \frac{{M{L^2}{T^{ - 2}}}}{{{T^1}}} = \frac{{M{L^2}}}{{{T^3}}}\)

∴ The dimensional formula of power P is [ML²T^-3].

Concept:

• Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit and Dimension is a mathematical tool used for studying the nature of physical quantities. 
• The basic concept of dimensions is that we can add or subtract only those quantities which have the same dimensions.
• And the dimensional formula is defined as the expression of the physical quantity in terms of mass, length, and time.

Explanation:

Power – It is defined as the rate of doing work.

\(\therefore P = \frac{W}{t}\)

Where, P = power, W = work done and t = time.

Now,

As Work done = Force × Distance

Force = mass × acceleration

∴ Dimensional formula of force (F) = [M] × [LT^-2] = [MLT^-2]

Dimensional formula of work (W) = [ML²T^-2]

Dimensional formula of time (t) = [T¹]

\(P = \frac{{M{L^2}{T^{ - 2}}}}{{{T^1}}} = \frac{{M{L^2}}}{{{T^3}}}\)

∴ The dimensional formula of power P is [ML²T^-3].

Concept:

Capacitance

• Capacitance is a property of the electric conductor measured by the amount of separated electric charge that can be stored on it per unit change in electrical potential.
• In capacitor, amount of charge, Q = CV, where, C = Capacitance, V = electrical potential
• A capacitance in an electric circuit is introduced by a device called a capacitor.
• The SI unit of the capacitance is Farad and is denoted by the F.

Explanation:

The SI unit of capacitance is Farad denoted by F.

Additional Information

• Resistance
  • ​​Resistance in an electric circuit is introduced by a device called a resistor.
  • Formula, V = IR, where I = electric current flowing in the circuit, V =  supplied voltage, R = Resistance
  • The SI unit of resistance is Ohm denoted by Ω.
• Conductance
  • ​The reciprocal of the resistance In a conductor is called conductance.
  • Relation, \(Conductance=\frac{1}{Resistance}\)
  • The SI unit of conductance is Ohm^-1.
• Inductance
  • ​The inductance in an electric circuit is introduced by a device called an inductor. 
  • The SI unit of the inductance is Henery denoted by H.