Units and Measurements
Understanding Quantities and Units
Quantities are physical properties that can be measured, such as:
Time
Mass
Temperature
Length
Units are standardized references used to measure quantities.
Different quantities require different units based on convenience:
Distance: centimeters, meters, kilometers, light-years
Time: seconds, minutes, hours, years
Types of Units
Fundamental Units
Basic units that cannot be derived from other units. There are 7 fundamental units:
Quantity Fundamental Unit
Mass kilogram (kg)
Time second (s)
Current ampere (A)
Length meter (m)
Amount of substance mole (mol)
Luminous intensity candela (cd)
Temperature kelvin (K)
Derived Units
,Units formed by combining fundamental units.
Examples:
Newton (N): kg ⋅ m ⋅ s −2
Joule (J): kg ⋅ m ⋅ s
2 −2
Minute: 60 seconds
Hour: 60 minutes
Dimensions
Dimensions represent the nature of physical quantities independent of units used.
Dimension Symbols:
Mass: M
Length: L
Time: T
Temperature: K
Current: I
Amount of substance: mol
Luminous intensity: cd
Finding Dimensions 🔍
To find dimensions of any quantity:
1. Write its formula
2. Replace units with dimension symbols
3. Simplify
Examples:
Speed:
Formula: speed = distance
time
Dimensions: [speed] = L
T
= L ⋅ T
−1
Momentum:
Formula: momentum = mass × velocity
Velocity dimensions: L ⋅ T −1
Momentum dimensions: M × L ⋅ T = M ⋅ L ⋅ T −1 −1
, Kinetic Energy:
Formula: K E = mv 1
2
2
Ignore constants (like ) 1
2
Mass: M
Velocity squared: (L ⋅ T ) = L −1 2 2
⋅ T
−2
Energy dimensions: M ⋅ L ⋅ T 2 −2
Important: All types of energy and work have same dimensions: M ⋅ L 2
⋅ T
−2
Force:
Formula: F = ma
Mass: M
Acceleration: L ⋅ T −2
Force dimensions: M ⋅ L ⋅ T −2
Refractive Index:
Formula: n = speed1
speed2
Both have same dimensions: L ⋅ T
−1
Result:
−1
L⋅T 0 0 0
= 1 = M L T
L⋅T −1
Dimensionless quantities have no units and dimensions, represented as M 0 0
L T
0
or
simply []
Key Points
1. Dimensions remain same regardless of units used
2. Constants have no dimensions (like , π, etc.) 1
2
3. All energy forms share same dimensions: M ⋅ L ⋅ T 2 −2
4. Dimensionless quantities have no units (refractive index, sine of angles)
Dimensionless Quantities 🎯
Refractive index
Trigonometric functions (sin, cos, tan)
Strain
Relative density
Understanding Quantities and Units
Quantities are physical properties that can be measured, such as:
Time
Mass
Temperature
Length
Units are standardized references used to measure quantities.
Different quantities require different units based on convenience:
Distance: centimeters, meters, kilometers, light-years
Time: seconds, minutes, hours, years
Types of Units
Fundamental Units
Basic units that cannot be derived from other units. There are 7 fundamental units:
Quantity Fundamental Unit
Mass kilogram (kg)
Time second (s)
Current ampere (A)
Length meter (m)
Amount of substance mole (mol)
Luminous intensity candela (cd)
Temperature kelvin (K)
Derived Units
,Units formed by combining fundamental units.
Examples:
Newton (N): kg ⋅ m ⋅ s −2
Joule (J): kg ⋅ m ⋅ s
2 −2
Minute: 60 seconds
Hour: 60 minutes
Dimensions
Dimensions represent the nature of physical quantities independent of units used.
Dimension Symbols:
Mass: M
Length: L
Time: T
Temperature: K
Current: I
Amount of substance: mol
Luminous intensity: cd
Finding Dimensions 🔍
To find dimensions of any quantity:
1. Write its formula
2. Replace units with dimension symbols
3. Simplify
Examples:
Speed:
Formula: speed = distance
time
Dimensions: [speed] = L
T
= L ⋅ T
−1
Momentum:
Formula: momentum = mass × velocity
Velocity dimensions: L ⋅ T −1
Momentum dimensions: M × L ⋅ T = M ⋅ L ⋅ T −1 −1
, Kinetic Energy:
Formula: K E = mv 1
2
2
Ignore constants (like ) 1
2
Mass: M
Velocity squared: (L ⋅ T ) = L −1 2 2
⋅ T
−2
Energy dimensions: M ⋅ L ⋅ T 2 −2
Important: All types of energy and work have same dimensions: M ⋅ L 2
⋅ T
−2
Force:
Formula: F = ma
Mass: M
Acceleration: L ⋅ T −2
Force dimensions: M ⋅ L ⋅ T −2
Refractive Index:
Formula: n = speed1
speed2
Both have same dimensions: L ⋅ T
−1
Result:
−1
L⋅T 0 0 0
= 1 = M L T
L⋅T −1
Dimensionless quantities have no units and dimensions, represented as M 0 0
L T
0
or
simply []
Key Points
1. Dimensions remain same regardless of units used
2. Constants have no dimensions (like , π, etc.) 1
2
3. All energy forms share same dimensions: M ⋅ L ⋅ T 2 −2
4. Dimensionless quantities have no units (refractive index, sine of angles)
Dimensionless Quantities 🎯
Refractive index
Trigonometric functions (sin, cos, tan)
Strain
Relative density
