Units and Measurement: Comprehensive NEET Physics Notes

1. Introduction

1.1 Understanding Measurement

Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called a unit. The result of a measurement is expressed by a number (or numerical measure) accompanied by a unit. Despite the large number of physical quantities, we only need a limited number of units since they are inter-related. Units for the fundamental or base quantities are called fundamental or base units, while units for all other physical quantities can be expressed as combinations of the base units, known as derived units. The complete set of these units is known as the system of units.

NEET Tip:

Understand the difference between base units and derived units, as this is fundamental for solving problems related to measurements.

2. The International System of Units (SI)

2.1 Historical Systems of Units

Previously, different systems of units were used by scientists in various countries. The CGS, FPS (British), and MKS systems were prominent:

• CGS: Centimetre, gram, second
• FPS: Foot, pound, second
• MKS: Metre, kilogram, second

2.2 The SI System

The Système Internationale d’Unités (SI) is the modern form of the metric system and is the most widely used system of measurement. Developed by the Bureau International des Poids et Mesures (BIPM), it was revised by the General Conference on Weights and Measures in 2018. The SI system is based on seven base units, shown in Table 1.1.

2.3 Supplementary Units

Besides the seven base units, there are two more units for plane angle (radian, symbol rad) and solid angle (steradian, symbol sr).

Did You Know?

The SI unit of time, the second, is defined by the caesium frequency, which is 9192631770 Hz. This is the vibration frequency of the caesium-133 atom.

3. Significant Figures

3.1 Definition and Importance

Significant figures in a measured quantity include all the digits that are known reliably plus the first uncertain digit. For instance, if the period of oscillation of a pendulum is 1.62 s, the digits 1 and 6 are certain, while 2 is uncertain.

3.2 Rules for Counting Significant Figures

1. All non-zero digits are significant.
2. All zeros between two non-zero digits are significant.
3. For numbers less than one, zeros to the right of a decimal point and to the left of the first non-zero digit are not significant.
4. Trailing zeros in a number without a decimal point are not significant.
5. Trailing zeros in a number with a decimal point are significant.

Common Misconception:

Trailing zeros are often misinterpreted in terms of significance. Remember, they are only significant if a decimal point is present.

3.3 Rounding Off

To round off numbers:

• If the digit to be dropped is more than 5, the preceding digit is increased by 1.
• If the digit to be dropped is less than 5, the preceding digit remains unchanged.
• If the digit to be dropped is exactly 5, the preceding digit is increased by 1 if it is odd, and remains unchanged if it is even.

NEET Problem-Solving Strategy:

When performing calculations, keep one more digit than the significant figures required and round off at the end to avoid intermediate rounding errors.

4. Dimensions of Physical Quantities

4.1 Understanding Dimensions

Dimensions of a physical quantity are the powers to which the base quantities are raised to represent that quantity. For example, the dimensions of force are $[ML{T}^{-2}]$.

4.2 Dimensional Formulae

Dimensional formulae express the dimensions of a physical quantity in terms of the base quantities. For example:

• Volume: $[M^0{L}^3{T}^0]$
• Speed: $[M^{0}L{T}^{-1}]$

Real-life Application:

Dimensional analysis helps in verifying the correctness of physical equations. If the dimensions on both sides of an equation are not the same, the equation is incorrect.

5. Dimensional Analysis and Its Applications

5.1 Checking Dimensional Consistency

Dimensional analysis is used to check the consistency of equations by ensuring that both sides have the same dimensions. This method cannot verify the numerical correctness of the equation but can indicate whether the equation is dimensionally correct.

5.2 Deducing Relations Among Physical Quantities

Using dimensional analysis, one can deduce the relationships between different physical quantities. For instance, the time period of a simple pendulum $T$ depends on its length $l$ and the acceleration due to gravity $g$: $T=2\pi \sqrt{\frac{l}{g}}$

NEET Exam Strategy:

Familiarize yourself with common dimensional formulas and practice using dimensional analysis to verify the correctness of equations. This technique can save time during the exam.

Stimulations:

Quick Recap

• Measurement involves comparing with a unit.
• SI units are the standard for scientific measurement.
• Significant figures reflect the precision of a measurement.
• Dimensions help in analyzing physical quantities.
• Dimensional analysis is a tool to check the consistency of equations and derive relations.

Practice Questions

1. What is the SI unit of luminous intensity?
2. Convert a speed of 90 km/h into m/s.
3. How many significant figures are in the measurement 0.00750 m?
4. Check the dimensional consistency of the equation for kinetic energy $E=\frac{1}{2}m{v}^2$.
5. A rectangle has length 5.6 m and width 3.2 m. Calculate its area and express it with the correct significant figures.

Solutions

1. The SI unit of luminous intensity is candela (cd).
2. Speed in m/s: $90\times \frac{1000}{3600}=25\text{ m/s}$.
3. The number of significant figures in 0.00750 m is 3.
4. Dimensions of $E$: $[M{L}^2{T}^{-2}]$; Dimensions of $\frac{1}{2}m{v}^2$: $[M{L}^2{T}^{-2}]$. The equation is dimensionally consistent.
5. Area: $5.6\times 3.2=17.92{\text{ m}}^2\to 17.9{\text{ m}}^2$ (three significant figures).

Glossary

• Base Unit: A fundamental unit that is defined arbitrarily and independently of other units.
• Derived Unit: A unit that is derived from the base units.
• Dimensional Analysis: A method to check the consistency of equations using dimensions.
• Significant Figures: Digits in a measurement that are known reliably plus the first uncertain digit.

Important Formulas from Units and Measurements

1. Physical Quantity and Measurement

• Physical Quantity: $Q=n\times u$

2. SI Base Units

• Length: $meter(m)$
• Mass: $kilogram(kg)$
• Time: $second(s)$
• Electric Current: $ampere(A)$
• Thermodynamic Temperature: $kelvin(K)$
• Amount of Substance: $mole(mol)$
• Luminous Intensity: $candela(cd)$

3. Derived Units

• Force (Newton): $F=m\times a$
• • SI Unit: $kg\cdot m/s^2$
• Energy (Joule): $E=F\times d$
• • SI Unit: $kg\cdot m^2/s^2$
• Power (Watt): $P=\frac{E}{t}$
• • SI Unit: $kg\cdot m^2/s^3$

4. Dimensional Formulas

• Force: $[ML{T}^{-2}]$
• Energy: $[M{L}^2{T}^{-2}]$
• Power: $[M{L}^2{T}^{-3}]$

5. Significant Figures

• For multiplication/division: The result should have as many significant figures as the value with the least significant figures.
• For addition/subtraction: The result should have as many decimal places as the value with the least decimal places.

6. Important Conversions

• 1 m = $10^{3}mm$
• 1 kg = $10^{3}g$
• 1 N = $10^{5}dynes$
• 1 J = $10^{7}ergs$