Unit and Dimensions
8.0 Dimensional formula
8.0 Dimensional formula
Dimensional formula: The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is known as the dimensional formula of the given physical quantity.
Consider the physical quantity force. Force can be represented as,
$${\text{Force}} = {\text{mass}} \times {\text{acceleration}}$$$${\text{Force}} = {\text{mass}} \times \frac{{{\text{length/time}}}}{{{\text{time}}}}$$$${\text{Force}} = {\text{mass}} \times {\text{length}} \times {\left( {{\text{time}}} \right)^{ - 2}}$$
So, the dimensions of force are $1$ in mass, $1$ in length and $-2$ in time.
Thus the dimension of the force can be represented as,
$$\left[ {{\text{Force}}} \right] = \left[ {ML{T^{ - 2}}} \right]$$where,
$M:$ Mass
$L:$ Length
$T:$ Time
Momentum can be represented as,
$${\text{Momentum}} = {\text{mass}} \times \frac{{{\text{length}}}}{{{\text{time}}}}$$$${\text{Momentum}} = {\text{mass}} \times {\text{length}} \times {\left( {{\text{time}}} \right)^{ - 1}}$$Thus the dimension of the momentum can be represented as,
$$\left[ {{\text{Momentum}}} \right] = \left[ {ML{T^{ - 1}}} \right]$$
Energy can be represented as,
$${\text{Energy}} = {\text{Force}} \times {\text{displacement}}$$$${\text{Energy}} = {\text{mass}} \times {\text{acceleration}} \times {\text{displacement}}$$$${\text{Energy}} = {\text{mass}} \times \frac{{{\text{velocity}}}}{{{\text{time}}}} \times {\text{displacement}}$$$${\text{Energy}} = {\text{mass}} \times \frac{{{\text{displacement/time}}}}{{{\text{time}}}} \times {\text{displacement}}$$$${\text{Energy}} = {\text{mass}} \times {\left( {{\text{displacement}}} \right)^2} \times {\left( {{\text{time}}} \right)^{ - 2}}$$Thus the dimension of the energy can be represented as,$$\left[ {{\text{Energy}}} \right] = \left[ {M{L^2}{T^{ - 2}}} \right]$$
