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A particle of mass m is located in a reg...

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.

Text Solution

Verified by Experts

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

  • The potential energy of a particle from a distance x from an origin, changes according to the formula U=(Asqrtx)/(x+B) where A and B are constant so the dimension of AB=……

    A
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    B
    `M^(1)L^(2)T^(-2)`
    C
    `M^(3/2)L^(3/2)T^(-2)`
    D
    `M^(1)L^(7/2)T^(-2)`

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