Notes

Units and Dimensions

Introduction to Units and Dimensions

In Physics, accurate measurement of physical quantities is crucial. To ensure consistency and standardization in measurements, we use units and dimensions. This chapter helps in understanding how to express physical quantities through units and analyze relationships between them using dimensional analysis.

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1. Physical Quantities

A physical quantity is any quantity that can be measured. For example, length, mass, time, and temperature are physical quantities.

Classification of Physical Quantities:

1. Fundamental Quantities: These are the basic quantities that are not derived from any other quantities. Examples:
   • Length (L)
   • Time (T)
   • Mass (M)
   • Electric Current (I)
2. Derived Quantities: These are derived from fundamental quantities. Examples:
   • Velocity = Length/Time (L/T)
   • Force = Mass × Acceleration (MLT⁻²)

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2. Units of Measurement

A unit is the standard by which a physical quantity is measured. The magnitude of a quantity is expressed as the product of the numerical value and the unit.

Types of Units:

1. Fundamental Units: The units of fundamental physical quantities such as meter (m) for length, kilogram (kg) for mass, and second (s) for time.
2. Derived Units: The units derived from fundamental units. Examples:
   • Velocity: meters per second (m/s)
   • Force: Newton (N) = kg·m/s²
   • Energy: Joule (J) = kg·m²/s²

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3. Systems of Units

Different systems of units are used worldwide. The most common systems are:

1. SI System (International System of Units): It is the most widely used system in scientific measurements. It consists of seven base units:
   • Meter (m) for length
   • Kilogram (kg) for mass
   • Second (s) for time
   • Ampere (A) for electric current
   • Kelvin (K) for temperature
   • Mole (mol) for the amount of substance
   • Candela (cd) for luminous intensity
2. CGS System: The units are based on centimeter, gram, and second.
   • Length: centimeter (cm)
   • Mass: gram (g)
   • Time: second (s)
3. FPS System: The units are based on foot, pound, and second. This system is used mostly in the United States.
   • Length: foot (ft)
   • Mass: pound (lb)
   • Time: second (s)

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4. Dimensions of Physical Quantities

Dimensions of a physical quantity represent its dependence on the fundamental units (length, mass, and time). A dimensional formula expresses a physical quantity in terms of the base quantities.

Dimensional Formula:

The dimensional formula of a physical quantity is the expression that shows how and which of the fundamental quantities are involved in the measurement of that quantity.

• For example:
  • Velocity = Distance/Time = [LT⁻¹]
  • Force = Mass × Acceleration = [MLT⁻²]
  • Energy = Force × Distance = [ML²T⁻²]

Dimensional Equation:

The equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation. For example:

• For Force: F=[MLT−2]F = [MLT⁻²]F=[MLT−2]

Dimensionless Quantities:

Some physical quantities have no dimensions, such as:

• Angle (measured in radians)
• Strain

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5. Dimensional Analysis

Dimensional analysis is a technique used to check the correctness of physical equations and derive relations between different physical quantities.

Applications of Dimensional Analysis:

1. Checking Dimensional Consistency: This ensures that both sides of an equation have the same dimensions. If the dimensions on both sides are the same, the equation is dimensionally correct.
   • Example: For the equation v2=u2+2asv^2 = u^2 + 2asv2=u2+2as, check the dimensions of each term:
     • v2v^2v2: [L2T−2][L^2T⁻²][L2T−2]
     • u2u^2u2: [L2T−2][L^2T⁻²][L2T−2]
     • 2as2as2as: [L2T−2][L^2T⁻²][L2T−2] Since the dimensions on both sides match, the equation is dimensionally correct.
2. Deriving Relationships: If we know how different quantities depend on each other dimensionally, we can derive their mathematical relationships.
   • Example: Deriving the formula for time period TTT of a simple pendulum:
     • TTT depends on length LLL and gravitational acceleration ggg.
     • By dimensional analysis: T=k×L/gT = k \times \sqrt{L/g}T=k×L/g​
3. Unit Conversion: Dimensional analysis helps in converting units from one system to another.
   • Example: Convert speed from km/h to m/s.
     • 1 km/h = 1000 m3600 s=518 m/s\frac{1000 \, \text{m}}{3600 \, \text{s}} = \frac{5}{18} \, \text{m/s}3600s1000m​=185​m/s

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6. Common Derived Quantities and their Dimensional Formulas

Physical Quantity	Formula	SI Unit	Dimensional Formula
Velocity	Distance/Time	m/s	[LT⁻¹]
Acceleration	Velocity/Time	m/s²	[LT⁻²]
Force	Mass × Acceleration	Newton (N)	[MLT⁻²]
Energy	Force × Distance	Joule (J)	[ML²T⁻²]
Power	Energy/Time	Watt (W)	[ML²T⁻³]
Pressure	Force/Area	Pascal (Pa)	[ML⁻¹T⁻²]

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7. Limitations of Dimensional Analysis

Though dimensional analysis is a powerful tool, it has certain limitations:

1. Cannot Determine Dimensionless Constants: Dimensional analysis cannot give the exact numerical value of dimensionless constants.
   • Example: In the equation v=2ghv = \sqrt{2gh}v=2gh​, the constant 2 cannot be derived by dimensional analysis.
2. Does Not Work for Trigonometric, Exponential, and Logarithmic Functions: Dimensional analysis is not applicable when the equation involves trigonometric, logarithmic, or exponential functions.
3. Cannot be Used to Derive Equations with Multiple Quantities: If the relationship between quantities is complex, involving multiple terms, dimensional analysis might not help.

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8. Important Questions for Practice

1. What is the dimensional formula of force and work?
2. Check whether the equation s=ut+12at2s = ut + \frac{1}{2}at^2s=ut+21​at2 is dimensionally correct.
3. How is the dimensional formula of pressure derived?
4. Convert a speed of 90 km/h to m/s.
5. Derive a relation between the time period of a pendulum and its length using dimensional analysis.

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Conclusion

Units and dimensions are the foundation of understanding and solving problems in Physics. Dimensional analysis, in particular, is a versatile tool that ensures the correctness of equations and aids in deriving formulas and converting units.

By mastering the concepts of units and dimensions, students can significantly improve their problem-solving skills in Physics.