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Dimension in Physics

What is Dimension in Physics

The dimension of a physical quantity shows the application of the method of fundamental quantities like Mass (M), Length (L), and Time (T) to relate the fundamental units of a physical quantity. In dimension, we derive quantities by applying physical quantities. To understand dimension, we need to understand the method of measurements, fundamental quantities, derived quantities, and even their corresponding units in physics.

Dimension in physics
A table of dimension in of quantities in Physics

This dimension helps us in finding the relationship between quantities. As we mentioned earlier, we apply Mass in the form of M, Length in the form of L, and Time in the form of T. Additionally, units like temperature will be in the form of K or θ, electric current in the form of (A) which is an expression for ampere, and finally luminous intensity as (II). Below are 2 videos explaining dimensions and their calculations.

Video 1

Video 2

Dimension in Physics

The Fundamental Dimensions

Physics identifies several base quantities, each with its own dimension:

• Length (L): Measures distance.

• Mass (M): Quantifies the amount of matter.

• Time (T): Tracks the progression of events.

• Electric Current (I): Represents the flow of electric charge.

• Thermodynamic Temperature (Θ): Indicates the thermal state.

• Amount of Substance (N): Counts entities like atoms or molecules.

• Luminous Intensity (J): Measures the perceived power of light.

These foundational dimensions serve as building blocks for all other physical quantities. 

Derived Dimensions and Their Significance

By combining base dimensions, we derive dimensions for other physical quantities. For example:

Velocity: Expressed as length per unit time (L/T), indicating how position changes over time.

• Force: Defined as mass times acceleration (ML/T²), representing the interaction that changes an object’s motion.

Energy: Calculated as force applied over a distance (ML²/T²), denoting the capacity to perform work.

Table of Dimensions in Physics

QUANTITYDIMENSION
SpeedLT-1
VelocityLT-1
AccelerationLT-2
ForceMLT-2
Tensile stressML-1T-2
Elastic ForceMLT-2
Young ModulusML-1T-2
Work done in elastic spring/stringML2T-2
WorkML2T-2
Work done horizontalML2T-2
Work done verticalML2T-2
Kinetic energyML2T-2
Potential energyML2T-2
PowerML2T-3
Projectile time of flightT
Projectile maximum heightL
Projectile rangeL
Moment of forceML2T-2
DensityML-3
UpthrustMLT-2
Centripetal accelerationLT-2
Centripetal forceMLT-2
Linear velocityLT-1
Linear accelerationLT-2
PeriodT
FrequencyT-1
Period of simple pendulumT
MomentumMLT-1
Impulse
PressureML-1T-2
Work done by expanding gasML2T-2
WavelengthM
Wave velocityLT-1
Gravitational ForceMLT-2
Gravitational potentialL2T-2
Escape velocityLT-1
Electric forceMLT-2
Magnetic forceMLT-2
Dimension Table in Physics

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How to Calculate Dimensions of Physical Quantity

To calculate the dimensions of a physical quantity, we need to be guided on the measurement of quantities in units of Length, Mass, and Time. These involve the dimensions Length (L), Mass (M), and Time (T) respectively. To find the dimension of any quantity, we substitute the units of the quantity into L, M, and T. You insert the symbols into a bracket in this way: [L], [M], and [T]. For example, to find the dimension of speed, we first write the formula as

Speed = Distance / Time

We know that the unit for Distance = meter (m) and meter is a length, which can be, Dimension of Distance = [L]

Note: We only have m which is [L]. If we have m2, we can now write [L2], and Time is in seconds which can also be T, Thus, the dimension of Time = T, To derive the dimension of speed, we say: Speed = Distance / Time. This implies that Speed = [L] / [T]. Therefore, the dimension of Speed = [LT-1] or [L][T-1].

Worked Examples on Dimension in Physics

Here are examples to make you understand dimension in physics: how to calculate dimensions of physical quantities

Example 1

  1. Find the dimension of an Area and Volume.

Solution:

For Area = Length x Breadth

Which shows that the dimension of Area = [L] x [L] = [L2]

Now, since volume = Length x Base x Height

The dimension of volume = [L] x [L] x [L] = [L3]

Example 2

2. Deduce the dimension of (a) force and (b) Pressure

Solution:

(a) force = mass x acceleration

which shows that

Dimension of force = dimension of mass x dimension of acceleration

and the Dimension of mass = [M]

while the Dimension of acceleration = Dimension of Velocity / Dimension of Time

and Dimension of Velocity = Dimension of Displacement [L] / Dimension of Time [T]

Thus, Dimension of Velocity = [L] / [T] = [L] [T-1] or [LT-1]

Therefore, by substituting the dimensions of velocity and time into acceleration,

we have

dimension of acceleration = [LT-1] / [T] = [LT-2]

Now, to finally derive the acceleration of force,

we say

Dimension of force = [M] x [LT-2] = [M] [LT-2] or [MLT-2]

(b) To find the dimension of pressure

We apply the formula pressure = force / area

which means Dimension of pressure = Dimension of force / Dimension of area

and Dimension of force = Dimension mass x Dimension of acceleration

you can remember that we earlier calculated the dimension of force as [MLT-2]

Also, Dimension of area = [L] x [L] = [L2]

Thus, Dimension of pressure = [MLT-2] / [L2]

This implies

Dimension of pressure = [ML-1T-2]

Example 3

Find the dimension of the following:

a. work

b. Velocity

c. Linear acceleration

d. Power

e. Pressure

f. Momentum

g. Impulse

Solution:

a. Work = Force x Distance

Dimension of Work = Dimension of Force x Dimension of Distance

Therefore, Dimension of Work = MLT-2 x L

This implies that, Dimension of work = ML2 T-2

b. Velocity = Displacement / Time = L / T = LT-1

c. Acceleration = Velocity / Time = LT-1 / T = LT-2

d. Power = work done / time = ML2 T-2 / T = ML2 T-3

e. Pressure = Force / Area = MLT-2 / L3 = ML-2T-2

f. Momentum = mass x velocity = M x LT-1 = MLT-1

g. Impulse = Force x Time = MLT-2 x T = MLT-1

Example 4

Find the dimension in the electrical system of the following:

a. Electric charge

b. Electrical Potential (voltage)

c. Capacitance (c)

Solution:

a. To find electric charge, we use the formula

electric charge = current (A) x time (T)

Thus, electric charge = [A] x [T] = [AT]

b. Electric potential (voltage) = power / current

Dimension voltage = ML2 T-3 / A = [MA-1L2 T-3]

c. Capacitance = charge / potential = [AT] / [MA-1L2 T-3]

Thus, capacitance = [M-1A2L-2 T4]

The Role of Dimensional Analysis

Dimensional analysis is a powerful tool in physics, allowing us to: 

Verify Equations: Ensuring both sides of a physical equation have the same dimensions confirms its validity. 

Derive Relationships: By analyzing the dimensions involved, we can deduce how different quantities depend on each other.

Convert Units: Facilitates transitioning between measurement systems by understanding the underlying dimensions.

Beyond the Observable: Higher Dimensions

While our everyday experience is confined to three spatial dimensions and one temporal dimension, theoretical physics ventures into higher-dimensional spaces:

• Fourth Dimension: Incorporates time with the three spatial dimensions, forming the fabric of spacetime in Einstein’s theory of relativity.

• Fifth Dimension and Beyond: Theories like Kaluza-Klein propose additional dimensions to unify fundamental forces, suggesting that these extra dimensions are compactified or “curled up” at scales beyond current detection. 

These higher dimensions, though not directly observable, provide profound insights into the fundamental nature of the universe.

Dimensions in Mathematics and Beyond

In mathematics, dimensions extend beyond physical interpretation:

• Fractal Dimensions: Describe complex, self-similar structures that don’t conform to traditional integer dimensions, offering insights into patterns found in nature.

• Abstract Spaces: Concepts like phase space in physics utilize higher-dimensional mathematical spaces to represent all possible states of a system.

These mathematical frameworks enrich our understanding and modeling of complex systems.

Reference

  1. Wikipedia