Understanding the Dimensions of Force in Physics

Force is a fundamental concept in physics, representing the interaction that causes an object to change its motion or state of rest. The dimensions of force describe its relation to the fundamental physical quantities—mass, length, and time—using dimensional analysis.

Definition and Dimensional Formula of Force

Force is defined as the product of mass and acceleration. The SI unit of force is the newton (N). The dimensional formula provides insight into the fundamental quantities involved in the physical quantity of force.

Mathematically, force ($F$) is given by $F = m \times a$, where $m$ is mass and $a$ is acceleration. The dimensional formula for force is $[M L T^{-2}]$.

Derivation of the Dimensions of Force

The dimensional derivation begins with Newton's second law, $F = m \times a$. Mass ($m$) has the dimension $[M]$. Acceleration ($a$) is the rate of change of velocity, which is length per unit time squared, or $[L T^{-2}]$.

Combining these, force is expressed dimensionally as:

$[M] \times [L T^{-2}] = [M L T^{-2}]$

Units of Force in Different Measurement Systems

Various measurement systems assign different names and magnitudes to the unit of force. The fundamental systems are CGS, MKS, FPS, and SI. Each uses different base units for mass, length, and time.

Dimensional Analysis and Applications

Dimensional analysis helps to verify the correctness of physical equations and to convert units between measurement systems. The dimensional formula for force is often used to derive the dimensions of other related quantities such as momentum, energy, and pressure.

The relationship between force and pressure is given by pressure equals force divided by area, so the dimensions of pressure are $[M L^{-1} T^{-2}]$. For further reference, see Dimensions Of Stress.

Examples of Dimensional Formulas Related to Force

Several important physical quantities are derived from force, and their dimensional formulas can be evaluated using the dimensional formula of force.

• Work: $[M L^2 T^{-2}]$
• Energy: $[M L^2 T^{-2}]$
• Momentum: $[M L T^{-1}]$
• Impulse: $[M L T^{-1}]$
• Viscosity: $[M L^{-1} T^{-1}]$

The calculation of these related quantities can be explored in more detail in the article on Dimensions Of Work.

Seven Fundamental Quantities and Force

The dimensional formula of force involves three of the seven fundamental quantities—mass, length, and time. These fundamental quantities are the basis for all physical quantities, both fundamental and derived.

A solid understanding of forces and their dimensions is essential for analyzing many problems in classical mechanics. For example, quantities like linear momentum and speed have dimensions closely related to force. For further reading, refer to Dimensions Of Linear Momentum and Dimensions Of Speed.

Dimensional Formulas of Force-Related Quantities

Other physical quantities closely related to force include force constant and density. The force constant, used in the context of springs, has the dimensional formula $[M T^{-2}]$. Density, which is mass per unit volume, has the formula $[M L^{-3}]$.

A detailed discussion on the dimensions of density can be found at Dimensions Of Density.

Coefficient of Viscosity and Force

The coefficient of viscosity is defined using force, distance, area, and velocity. Its dimensional formula is $[M L^{-1} T^{-1}]$, which arises from $\dfrac{\text{Force} \times \text{Distance}}{\text{Area} \times \text{Velocity}}$.

For further insights into viscosity and its dimensional analysis, visit Dimensions Of Viscosity.