Dimensions of Mass and Density When Force, Length, and Time Are Considered as Base Quantities

In traditional physics, mass and density are often derived from the definitions of force, length, and time as base quantities. However, when considering these quantities as the fundamental base units, the dimensional formulas of mass and density can be derived through a series of conversions and applications of dimensional analysis. Let's explore the process step-by-step.

Force (F), Mass (M), and Time (T) as Base Quantities

Typically, force, mass, and time are not base quantities but rather derived from the three base quantities: length, mass, and time. However, if we hypothetically consider force, length, and time as base quantities, then the dimensional formulas of mass and density would differ.

Dimensional Formula for Mass (M)

From Newton's Second Law, we have: Force (F) Mass (M) * Acceleration (a) The dimensional formula of force (F) can be written as: F [ML^2T^(-2)] Given that acceleration (a) has the dimensional formula of: a [LT^(-2)] To find the dimensional formula for mass (M), we rearrange the equation: M F / a Substituting the dimensional formula of force and acceleration: M [ML^2T^(-2)] / [LT^(-2)] [ML^1T^(0)] [ML]

Dimensional Formula for Density (ρ)

Density (ρ) is defined as mass per unit volume: ρ Mass / Volume The dimensional formula for volume (V) can be expressed as: V Length^3 [L^3] Substituting the dimensional formula for mass and volume: ρ [ML] / [L^3] [M / L^2]

Alternative Method: Deriving Density Using Force

To further illustrate the dimensional formulas for mass and density, we can use the relationship between force, mass, and acceleration. Consider the following:

Dimensional Formula for Force (F)

Force (F) Newtons (N) Note that: 1 N 1 kg*m/s^2

Deriving Mass (M) from Force (F) and Acceleration (a)

F M*a Given that acceleration (a) is the second derivative of distance with respect to time, we can express it as: a [L / T^2] Substituting into the equation: F M * [L / T^2] Rearranging to solve for mass (M): M F * [T^2 / L]

Dimensional Formula for Density (ρ)

ρ M / V [ML^(-2)T^2 / L^3] FL^(-4)T^2

Conclusion

When force (F), length (L), and time (T) are considered as base quantities, the dimensional formulas for mass (M) and density (ρ) are:

Mass (M) [FL^(-1)T^2] Density (ρ) [FL^(-4)T^2]

This illustrates the importance of consistent base units in physics and the intricate relationships between fundamental quantities. Understanding these relationships is crucial for accurate dimensional analysis and problem-solving in physics.

Keywords: mass dimensions, density dimensions, dimensional analysis

References:

Newton's Second Law (F ma) Derivative of distance with respect to time Dimensional analysis in physics