A particle of mass m is executing oscillation about the origin on X- axis Its potential energy is V(x)=kIxI Where K is a positive constant If the amplitude oscillation is a, then its time period T is proportional

Options

A. proportional to`1//sqrta`
B. independent of a
C. proportional tosqrta
D. proportional to `a^(3//2)`

Answer

A

Explanation

`V(x)=K|x|^(3) therefore [k]=([V])/([x]^(3))=(ML^(2)T^(2))/(L^(3))=ML^(-1)T^(-2 )`
now time period T may depend on `(mass)^(x)(amplitude)^(y)(K)^(z)`
therefore `[M^(0)L^(0)T][M]^(x)[L]^(y)[ML^(-1)T^(2)]^(x)=[M^(x+y)L^(y-x)T^(-2x)]`
Equating the powers, we get
`-2z=1` or `z=-1//2`
`y-z=0` `y=z=-1//2`
Hence `T prop ("amplitude")-1//2` or `T prop(1)/(sqrta)`