Which of the following is true for the mechanical equivalent of...

According to the relation between work and heat, $W = JH$, where $W$ is work, $H$ is heat, and $J$ is the mechanical equivalent of heat.

Rearranging the formula, $J = \dfrac{W}{H}$.

Since both work and heat are forms of energy, their dimensional formulas are the same, which is $[M L^2 T^{-2}]$.

Substituting the dimensions, $[J] = \dfrac{[M L^2 T^{-2}]}{[M L^2 T^{-2}]} = [M^0 L^0 T^0]$.

Therefore, the mechanical equivalent of heat is a dimensionless quantity.