Dimensional Equations

Dimensions are powers to which fundamental units are raised in derived units. It helps first identify which fundamental units help derive which derived units and the degrees to which fundamental units are raised. The dimensional formula helps us derive the dimensional equations for various units.

Dimensions are also important in geometry. The equations of various shapes change in one dimension, two dimensions, or in higher dimensions. The distance formula is used to form such equations of shapes. The dimensional analysis is used to identify the nature of relationships between physical quantities through the fundamental units which derive them.

Dimensional Equation

The differential equations for any physical quantity are derived with the help of the dimensional formula. We can also study the circle equation with the help of dimensions. The dimensional formula is stated below.

Suppose Q is a physical quantity, the unit’s dimensional equation will be as follows:

Q = MaLbTc

This is the general dimensional equation. 

Here M represents the mass of the physical quantity, T represents the time aspect of the physical quantity, and L is the length of the physical quantity.

The a, b, and c are the powers to which M, L, and T are raised, respectively.

The dimensional equation of any physical quantity contains the dimensional formula for that physical quantity. The dementia formula on the right side is equated to the left-hand side of the equation, and the dimensional equation of physical quantity is obtained.

Let us look at dimensional equations of a few physical quantities.

• Acceleration

The acceleration of any object is a change in its velocity with respect to Time or acceleration due to gravitation. The unit of gravitation is ms-2. Here the length is raised to power 1, and Time is raised to power -2.

Hence, the dimension equation of acceleration will become L1M0T-2

which equates to L1T-2.

• Angular Displacement

The angular displacement is measured in radians. It cannot be represented in the form of fundamental units such as Time, length, and mass. Hence, the fireworks off Time and months will be all equal to 0. Hence, the dimensional equation of angular displacement becomes M0L0T0.

• The Angular Impulse

Angular impulse is made up of a fundamental quantity and a derived quantity. The formula for angular impulse is torque multiplied by Time. Hence, the unit of angular impulses is Nms. 

• The Coefficient of Surface Tension

The coefficient of surface tension is equal to four divided by the length. The Who is measured in newtons and length is measured in metres. Hence, the Time will be raised to the power of -2 and must be raised to 1.

Hence the dimensional equation of the coefficient of surface tension will be M1L0T-2; hence it will become M1T-2.

Dimensional Analysis

In mathematical terms, the form of dimensional analysis is an analysis of what relationships exist between physical quantities. 

The relationship can be determined by the following term.

The units such as length, electric current, Time, and mass exist in the dimensional equation of the physical quantity.

The power to which these fundamental units are raised.

For example, let’s consider acceleration. The dimensional equation for acceleration in M0 L1 T-2.

From the above equation, we can perform the dimensional analysis.

The decoration tells us that the acceleration has no mass dimension and has Length and time dimensions.

The length is raised to 1, so acceleration is directly proportional to the length, and the time is raised to -2, the acceleration is inversely proportional to Time.

Relationships between Physical quantities with Dimensional Analysis

We established that the dimensional equation of acceleration is M0 L1 T-2.

Now let’s take a look at the dimensional equation of velocity.

The dimensional equation of velocity is M0 L1 T-1.

Upon comparison, We can identify that the power of mass is the same in both equations, and the power of length is similar.

However, Time is raised to a power of -1 in velocity while it is raised to the power of -2 in acceleration.

It tells us that when you divide velocity by Time, you get an acceleration of an object. Hence, the relationship between acceleration and velocity was identified with the help of dimensional equations of acceleration and velocity.

The Circle Equation

In the equation of a circle, the radius of the circle is considered. It is just a length quantity, and mass and Time are not involved.

The equation of the circle according to the distance formula is given below:

d = √( (x2 – x1)2 + (y2 – y1)2 )

By substitution and squaring,

d = (x – h)2 + (y -k)2

Here, the d represents the radius of the circle.

Such is the circle equation.

Conclusion

The dimensional formula helps in the revision of the dimensional equations. The dimensional equations represent what fundamental quantities make of the derived quantities and the powers to which they should be raised. Dimensional analysis is an analysis of these dimensional equations.

Through dimensional analysis, we can establish the relationships between two physical quantities by taking a look at the powers of fundamental units. However, there are certain drawbacks to dimensional analysis. It does not tell us whether a certain physical quantity is a vector quantity or a scalar quantity, and We don’t get any information about the dimensional constant from dimensional analysis.