Question

If all the terms in an equation have same units, is it necessary that they have same dimensions? If all the terms in an equation have same dimensions, is it necessary that they have same units?

Answer 1

Units can vary.

Explanation:

Yes, it is necessary that it has the same dimensions and no it is not necessary that all same dimensions have the same units because units can vary, it can be m/cm/ft etc. for length

For example,  

In the equation W = T + mv²

T = torque      

V = velocity

m = mass

Dimensional formula of each term in the equation is [M¹L²$T^-^2$] but the unit of work is joule , torque is newton- meter so here we concluded that the units are different.

Therefore, units are different.