The dimensional formula of the joule is [ML2T-2].
Here's a breakdown of why:
The joule (J) is the SI unit of energy and work. Since energy and work are fundamentally related (work is the transfer of energy), they have the same dimensional formula. We can derive this formula by analyzing the formula for kinetic energy, potential energy, or work itself.
Let's consider the formula for kinetic energy (KE):
KE = (1/2) mass velocity2
Dimensionally:
- Mass (m) has the dimension [M]
- Velocity (v) has the dimension [LT-1] (length per time)
Therefore, the dimensional formula for kinetic energy (and thus energy and work/joule) is:
[M (LT^{-1})^2] = [ML^2T^{-2}]
We can also arrive at this conclusion by considering the definition of work:
Work = Force x Distance
Dimensionally:
- Force (F) has dimensions [MLT-2] (from F=ma, mass x acceleration)
- Distance (d) has dimensions [L]
Therefore, the dimensional formula for work (and thus energy/joule) is:
[MLT^{-2} \cdot L] = [ML^2T^{-2}]
Therefore, regardless of whether we analyze it through kinetic energy, potential energy, or work, the dimensional formula for the joule remains [ML2T-2].
