Question

If Velocity `(V) , force(F) , and time (T)` are chosen as fundamental quantities , express `(a) mass and (b)` energy in terms of `V,F, and T`.

Answer 1

Let M = (some number ) `(V)^(a) (F)^(b) (T)^(c )`
Equating dimensions of both the sides , we get
`M^(1) L^(0) T^(0) = (1) [L^(1) T^(-1)]^(a) [M^(1) L^(1) T^(-2) ]^(b) [T^(1)]^(c )`
`= M^(b) L^( a+b) T^(a- 2b + c )`
Get `a = -1 , b = 1 , c = 1`.
M = ( some number) `(V^(-1) F^(1) T^(1)) [ M] = [ V^(-1) F^(1) T^(1)]`
Similarly , we can also express energy in terms of `V, F, and T`.
Let `[E]` = [ Some number] `[V]^(a) [F]^(b) [T]^( c )`
`rArr [ ML^(2) T^(-2)] = [ M^(0) L^(0) T^(0)] [ LT^(-1)]^(a) [ MLT^(-2)]^(b) [T]^(c)`
`rArr [M^(1) L^(2) T^(2)] = [M^(0) L^(0) T^(0)] [ LT^(-1)]^(a) [MLT^(-2)]^(b) [T]^(c )`
`rArr [M ^(1)L^(2) T^(2)] = [ M^(b) L^( a+ b + c)T^(-a - 2b+ c )]`
rArr `1 = b , 2 = a + b+ c , -2 = -a -2b + c `
Get `a = 1 , b = 1 , c = 1`.
`rArr E =` ( some number) `V^(1) F^(1) T^(1) or [E] = [ V^(1) [F^(1)] [T^(1)]`.