Question 5: What does dimensions of a physical quantity mean? Explain what are its applications and limitation.

### Definition of dimensions

The powers (positive or negative) to which the fundamental units of first system must be raised to give the unit of a given physical quantity (derived units) are called dimensions of that quantity.

OR

Equivalently, it is the number of times the fundamental units (mass, length, time etc) appear in the derived physical quantity.

Dimensions relate the derived quantity to the base quantities. It enables us to know how a (derived) physical quantity is related to the base units of length, mass, time and other base quantities. Dimensions are represented by capital letters enclosed in squared brackets.

#### Dimensionsof three base quantities are;

1. Dimension of length is [L].
2. Dimension of mass is [M].
3. Dimension of time is [T].

Dimensions of derived quantities are the products of the dimensions of the base quantities from which they are derived. Let’s take some examples:

1) Area of a square is obtained by multiplying the two of its sides. Therefore, the unit of area is m2. Replace the unit of length by the corresponding dimensions, we have dimensions of area = [L] [L] = [L2].

2) Unit of acceleration is given by ms-2. Replace the base units (meter and second) by the corresponding dimensions, we have dimensions of acceleration = [L] [T-2] = [LT-2].

3) Unit of force is N = kg m s-2. So its dimensions will be [M] [L] [T-2] = [MLT-2].

### Applications of dimensions

• Only quantities of the same dimensions can be mathematically operated (added, subtracted, multiplied, divided etc.) (For example, we cannot add 2 kg of sugar with 10 seconds of time because both have different dimensions) therefore, in an equation both sides must have same dimensions. So dimensions can be used to check the correctness of an equation.
• Dimensional analysis can be used to derive the possible formula of a physical quantity.
• To derive equations showing the relation between different physical quantities.

### Limitations of dimensions

Although dimensional analysis is very useful, it cannot help us more due to the following reasons:

• Dimensionless quantities cannot be determined by this method.
• Physical quantities represented by a large number of base quantities are difficult to handle with this method.
• If there are addition and subtraction appearing on one side of the equation, we cannot apply this method.
• This is not applicable to logarithmic, trigonometric or exponential functions.