A particle of mass (m) is executing oscillations about the origin on the (x) axis. Its potential energy is `V(x) = k|x|^3` where (k) is a positive constant. If the amplitude of oscillation is a, then its time period (T) is.

Options

A. Proportional to `(1)/(sqrtA)`
B. Independent of A
C. Proportional to `sqrtA`
D. Proportional to `A^(3//2)`

Answer

A

Explanation

`U(x)=k|x|^(3)`
`k=(U)/(|x^(3)|)=(ML^(2)T^(2))/(L^(3))=ML^(-1)T^(-2)`
Now, `Tprop["Mass"]^(x)["Amplitude"]^(y)[k]^(z)`
`[T]=[M]^(x)[L]^(y)[ML^(-1)T^(-2)]^(z)`
`implies [T]=[M^(x+z)L^(y-z)[T^(-2z)]`
`-2z=1implies z=-1//2`
`y-z=0 implies y=z=-1//2`
Hence, `Tprop("Amplitude")^(-1//2)`
`implies Tprop(1)/(sqrtA)`