In two systems of relations among velocity , acceleration , and force are , respectively , `v_(2) = (alpha^(2))/( beta) v_(1) , a_(2) = alpha beta a_(1), and F_(2) = (F_(1))/( alpha beta)`. If `alpha and beta` are constants , then make relations among mass , length , and time in two systems.

Answer

`M_(2) = (M)/(alpha^(2) B^(2)) ; L_(2) = L_(1) (alpha^(3))/(beta^(3)) ; T_(2) = T_(1) (alpha)/(beta ^(2))`

Explanation

`v_(2) = v_(1) (alpha^(2))/( beta) rArr [L_(2) T_(2)^(-1)] = [L_(1) T_(1)^(-1)] ( alpha^(2))/( beta)` …..(i)
`a_(2) = a_(1) alpha beta rArr [L_(2) T_(2)^(-2)] = [L_(1) T_(1)^(-2)] alpha beta` ……(ii)
and `F_(2) = (F_(1))/( alpha beta) rArr [M_(2)L_(2) T_(2)^(-2)] xx (1)/( alpha beta)` ......(iii)
Dividing Eq. (iii) by Eq. (ii) , we get
`M_(2) = (M_(1))/((alpha beta) alpha beta) = (M_(1)) /(alpha^(2) B^(2))`
Squaring Eq. (i) by Eq. (ii) , we get `T_(2) = T_(1) (alpha)/(beta^(2))`