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Finding Dimensional Formula For Acceleration

By Kathleen Knowles, 10 Sep 2020

Physical quantities are used to quantify the properties of a system. To give meaning to these physical quantities, a numerical value and a unit, which is universally accepted, are combined. For example, to measure the length of a beam, we say the beam is 6 [numerical quantity] + meters [unit].

Dimensional formulas represent an essential way of expressing these systems in easy-to-understand formats.

Physical Quantities

To fully grasp the concept of dimensional formulas, we need to understand the classes of physical quantities. Generally, there are two classes of physical quantities: fundamental or basic quantities and derived quantities.

Fundamental or basic quantities are those quantities that are not defined using other physical quantities. Instead, they form a basis for which other quantities, referred to as derived, are defined. Basic quantities and their SI units include: length (metre (m)), mass (kilogramme (kg)), time (second (s)), temperature (Kelvin (K)), electric current (Amperes (A)), luminous intensity (candela (cd)), and amount of substance (mole (mol)). We are focusing on length, mass, and time in this article.

Applying Dimensional Formulas

The application of dimensional formulas in defining derived units has a few advantages

  • It demonstrates how other physical quantities can be represented in terms of their base units.
  • It acts as a check for dimensional correctness of a derived quantity formula.
  •  It reduces errors that occur when converting from one unit to another across different systems, (say from SI units to British units).

When a physical quantity is written as its dimensional formula, we obtain a dimensional equation.

To express dimensional formulas using length, mass, and time, we assign base values: [L] for length, [M] for mass, and [T] for time.

Prerequisites for the Dimensional Formula for Acceleration

To derive the dimensional formula for acceleration, we have to obtain the formulas for displacement and velocity, two quantities that acceleration is dependent on.

Dimensional Formula for Displacement

Displacement [d] is a change in position or movement. Length represents displacement.

[d] = [L]

This is also written as:

[d] = [M0 L1 T0]

This means that displacement has one dimension in length and no dimension in mass and time.

The dimensional formula for displacement is [L].

Dimensional Formula for Velocity

Velocity [v] is defined as a displacement with respect to time. This is represented as:

v = Displacement/Time

Since displacement [d] is given as [L] we have:

[v] = [L]/[T]

This can be further simplified as:

[v] = [L][T-1] or [LT-1]

This can be further expressed as:

[v] = [M0 L1 T-1]

The dimensional equation above is similar to the dimensional formula for displacement. Velocity has one dimension in length, minus one (-1) dimension in time, and no dimension in mass.

Dimensional Formula for Acceleration

Acceleration [a] can be defined as the rate of change of velocity with time. This can be represented as:

[a] = Change in velocity/Change in time

Previously, we derived velocity as [LT-1], therefore acceleration is given as:

[a] = [LT-1]/[T]

This can be further simplified as:

[a] = [L][T-2] or [LT-2]

This can be further expressed as:

[a] = [M0 L1 T-2]

The right side of the equation above shows the dimensional formula for acceleration. According to this dimensional equation, acceleration has one dimension in length, minus two (-2) dimensions in time, and no dimension in mass.

From the examples above, it is easy to attach units to various quantities. For example, acceleration can be expressed in meter per second squared (m.s-2) or feet per second squared (ft.s-2), while velocity can be expressed in meter per second (m.s-1) or kilometer per hour (km.hr-1). Whichever unit you wish to use, you can check its consistency with the dimensional formula.

To illustrate better, let us derive the dimensional formula for force.

Dimensional Formula for Force

Force [f] is defined as an impact that causes an object of a certain mass to accelerate. This can be put simply as

[f] = Mass * Acceleration

Using the dimensional formula for acceleration, we have:

[f] = [M] * [M0 L1 T-2]

This can be combined to give:

[f] = [M1 L1 T-2]

Force has one dimension in mass, one dimension in length, and less than two (-2) dimensions in time. Force is expressed in Newtons, kilogram meter per second squared (kg.m.s-2).

Other Dimensional Formulas

As highlighted earlier, one of the advantages of dimensional formulas is the ability to check for dimensional correctness in formulas. This means that with more and more complex formulas, these concepts can be applied. Other useful dimensional formulas and their equations include:

Power [P] = [M1 L2 T-3]

Density [D] = [M1 L3 T0]

Pressure [P] = [M1 L-1 T-2]

Energy [E] = [M1 L2 T-2]

Limitations of Dimensional Formulas

The use of dimensional formulas, however, has some limitations.

  • It is challenging to define functions when subjected to logarithmic or exponential functions.
  • Dimensional formulas cannot be used to calculate proportionality constants.
  • For more complex physical quantities, you may require the knowledge of other dimension formulas.
  • There are no unique letters used to differentiate between physical quantities. For example, Power [P] and Pressure [P]

Generally, to represent the dimensional formula for a physical quantity, the base quantities (Mass, Length, Time, etc.) need to be combined with their respective powers (0, 1, 2, etc.). Physical quantities do not necessarily need to have all the base quantities. Displacement [L], velocity [LT-1], and acceleration [LT-2] are good examples of such physical quantities.

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