Dimensions and Units of Work

The dimensional formula of work is \([ML^2T^{-2}]\). Its absolute unit in SI is joule and is denoted by \(J\). In the cgs system, the absolute unit of work is erg. From the relation

\[
W = FS, \quad (\text{when } \theta = 0^\circ)
\]

\[
1 \text{ joule} = 1 \text{ newton} \times 1 \text{ metre}
\]

Therefore, work done is said to be 1 joule, if a force of 1 newton displaces a body through 1 meter in the direction of force.

Again, from the equation \( W = FS \), we have

\[
1 \text{ erg} = 1 \text{ dyne} \times 1 \text{ cm}
\]

Work done is said to be 1 erg, if a force of 1 dyne displaces a body through 1 cm in the direction of force.

Relation between Joule and Erg

\[
1J = 1N \times 1m = 10^5 \text{ dyne} \times 100 \text{ cm}
\]

or

\[
1J = 10^7 \text{ erg}
\]

The gravitational unit of work in SI is \( \text{kg m} \) and in the cgs system, the unit is \( \text{g cm} \).

Work done is said to be 1 \( \text{kg m} \), if a force of 1 kgf displaces a body through 1 m in the direction of force.

\[
1 \text{kg m} = 1 \text{kgf} \times 1 \text{m} = 9.8J
\]

Work done is said to be 1 \( \text{g cm} \), if a force of 1 gf displaces a body through 1 cm in the direction of force.

\[
1 \text{g cm} = 1 \text{gf} \times 1 \text{cm} = 980 \text{erg}
\]