Introduction to Units and Dimensions
In Physics, every measurable physical quantity requires a standard for its measurement, known as a unit. Units and dimensions are essential tools that help scientists express, analyze, and compare physical quantities in a consistent manner. Through this chapter, we will explore the importance of units, the classification of physical quantities, and how dimensional analysis assists in checking the validity of equations.
1. Physical Quantities
A physical quantity is any property of a material or system that can be measured. Physical quantities can be broadly categorized into two types:
Types of Physical Quantities:
- Fundamental (Base) Quantities: These quantities are independent of any other physical quantities. They form the foundation for the measurement of other quantities. The seven fundamental quantities in the SI system are:
- Length (L)
- Mass (M)
- Time (T)
- Electric Current (I)
- Temperature (Θ)
- Amount of Substance (N)
- Luminous Intensity (J)
- Derived Quantities: Derived quantities are obtained from the fundamental quantities through mathematical relationships. For example:
- Velocity = Length/Time (L/T)
- Force = Mass × Acceleration (MLT⁻²)
- Energy = Force × Distance (ML²T⁻²)
Units of Measurement
A unit is a defined standard by which a physical quantity is measured. The magnitude of a physical quantity is expressed as the product of the numerical value and its unit.
Types of Units:
- Fundamental Units: These are the units of fundamental physical quantities, such as meter (m) for length, kilogram (kg) for mass, and second (s) for time.
- Derived Units: These are units derived from fundamental units through algebraic relationships. Some examples are:
- Velocity: meters per second (m/s)
- Force: Newton (N) = kg·m/s²
- Work or Energy: Joule (J) = kg·m²/s²
- Power: Watt (W) = Joule per second (J/s) = kg·m²/s³
Systems of Units
Different systems of units are used in different regions and fields. Some common systems include:
- SI System (International System of Units): The SI system is the most widely accepted system of measurement. It consists of seven base units:
- Length: Meter (m)
- Mass: Kilogram (kg)
- Time: Second (s)
- Electric Current: Ampere (A)
- Temperature: Kelvin (K)
- Amount of Substance: Mole (mol)
- Luminous Intensity: Candela (cd)
- CGS System: The CGS system is based on the centimeter, gram, and second as the base units. It is primarily used in small-scale scientific work.
- Length: Centimeter (cm)
- Mass: Gram (g)
- Time: Second (s)
- FPS System: The FPS system is used mostly in the United States. It is based on foot, pound, and second.
- Length: Foot (ft)
- Mass: Pound (lb)
- Time: Second (s)
4. Dimensions of Physical Quantities
The dimensions of a physical quantity indicate how it depends on the fundamental quantities like length, mass, and time. The dimensional formula is an expression that shows the power of the fundamental units in a derived unit.
Dimensional Formula:
The dimensional formula of a physical quantity is represented in terms of the base quantities (length, mass, time, etc.). For example:
- Velocity = Distance/Time = [LT⁻¹]
- Force = Mass × Acceleration = [MLT⁻²]
- Energy = Force × Distance = [ML²T⁻²]
Dimensional Equation:
The dimensional equation is obtained by equating a physical quantity with its dimensional formula. For example:
- For Force: F=[MLT−2]F = [MLT⁻²]F=[MLT−2]
- For Energy: E=[ML2T−2]E = [ML²T⁻²]E=[ML2T−2]
km/h) to meters per second (m/s), you can use the dimensional relationship between distance and time to correctly perform the conversion.
Dimensionless Quantities:
Some physical quantities have no dimensions and are called dimensionless quantities. These include:
- Pure numbers, such as angles (measured in radians or degrees)
- Strain (a ratio of two similar quantities, e.g., change in length to original length)
Dimensional Analysis
Dimensional analysis is the technique of checking equations or deriving relationships between physical quantities by comparing their dimensional formulas.
Applications of Dimensional Analysis
- Checking the Dimensional Consistency of Equations:
An equation is dimensionally consistent if the dimensions on both sides of the equation are the same. For example, consider the equation for uniformly accelerated motion. The terms in this equation (velocity, initial velocity, acceleration, and distance) are checked by comparing their dimensions. If the dimensions of each term match, the equation is said to be dimensionally correct.
- Deriving Relationships Between Physical Quantities:
Dimensional analysis helps derive the form of a physical relationship between quantities. For instance, the time period of a simple pendulum depends on its length and the gravitational acceleration. Using dimensional analysis, you can derive an expression for the time period, which shows how it relates to these factors. However, the constant involved in the relationship cannot be determined through dimensional analysis alone.
- Unit Conversion:
Dimensional analysis assists in converting units from one system to another. For example, when converting velocity from kilometers per hour (km/h) to meters per second (m/s), you can use the dimensional relationship between distance and time to correctly perform the conversion.
Limitations of Dimensional Analysis
- Cannot Determine Dimensionless Constants:
Dimensional analysis cannot provide the numerical value of constants that are dimensionless. For example, in certain physical equations, a constant like “2” may appear, but dimensional analysis will not reveal this value.
- Does Not Work for Trigonometric, Exponential, or Logarithmic Functions:
Dimensional analysis does not apply to equations involving trigonometric, logarithmic, or exponential functions because these functions involve dimensionless quantities.
- Unable to Derive Complex Relationships:
Dimensional analysis may fail when there are multiple terms on either side of an equation with different dimensions, or when the relationship between quantities is more complex and involves more than simple multiplication or division.
Important Practice Questions
- What is the dimensional formula of force and work?
- Check whether the equation s=ut+12at2s = ut + \frac{1}{2}at^2s=ut+21at2 is dimensionally correct.
- How is the dimensional formula of pressure derived?
- Convert a speed of 100 km/h to m/s.
- Derive the relation between the time period of a pendulum and its length using dimensional analysis.
