Question

A particle of mass m is located in a region where its potential energy `[U(x)]` depends on the position x as potential Energy `[U(x)]=(1)/(x^2)-(b)/(x)` here a and b are positive constants…
(i) Write dimensional formula of a and b
(ii) If the time perios of oscillation which is calculated from above formula is stated by a student as `T=4piasqrt((ma)/(b^2))`, Check whether his answer is dimensionally correct.

Answer 1

(i) `[a]=[Ux^2]=ML^2T^-2L^2=ML^4T^-2`
`[b]=[Ux]=ML^2T^-2L=ML^3T^-2`
(ii) `T=4piasqrt((ma)/(b^2))=4pisqrt((ma^3)/(b^2))`
Dimension of RHS`=[4pisqrt((ma^3)/(b^2))=sqrt((MM^3L^(12)T^-6)/(M^2L^6T^-4))T`.
So, his answer is dimensionally incorrect.