The dimension of mass in terms of energy, velocity, force is

1.	The dimension of mass in terms of energy, velocity, force is
Answer» We have to express the dimension of mass in TERMS of energy $\sf{E,}$ velocity $\sf{v}$ and force $\sf{F.}$ Solution 1:- Let the dimension of mass in terms of energy, velocity and force is as follows, $\longrightarrow\sf{[m]=E^a\,v^b\,F^c\quad\quad\dots(1)}$ Taking dimensions of each, $\longrightarrow\sf{[m]=[E]^a\,[v]^b\,[F]^c}$ $\longrightarrow\sf{M=(ML^2T^{-<klux>2</klux>})^a(LT^{-1})^b(MLT^{-2})^c}$ $\longrightarrow\sf{M^1\,L^0\,T^0=M^{a+c}\,L^{2a+b+c}\,T^{-2a-b-2c}}$ Equating like POWERS, $\longrightarrow\sf{a+c=1}$ $\longrightarrow\sf{2a+b+c=0}$ $\longrightarrow\sf{-2a-b-2c=0}$ By SOLVING them we get, $\longrightarrow\sf{a=1}$ $\longrightarrow\sf{b=-2}$ $\longrightarrow\sf{c=0}$ Hence (1) becomes, $\longrightarrow\sf{\underline{\underline{[m]=E^1\,v^{-2}\,F^0}}}$ Solution 2:- We know that kinetic energy is also an energy and is given by, $\longrightarrow\sf{E=\dfrac{1}{2}mv^2}$ We take the dimensions, $\longrightarrow\sf{[E]=\left[m\right]\left[v\right]^2}$ From this we get the dimension of mass, $\longrightarrow\sf{[m]=\dfrac{\,[E]\,}{\,[v]^2}\,}$ $\longrightarrow\sf{[m]=[E][v]^{-2}}$ Finally, we get, $\longrightarrow\sf{\underline{\underline{[m]=E^1\,v^{-2}\,F^0}}}$

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We have to express the dimension of mass in TERMS of energy $\sf{E,}$ velocity $\sf{v}$ and force $\sf{F.}$