The frequency of vibrations of a mass `m` suspended from a spring of spring const. `k` is given by `v=cm^(x)k^(y),` where `c` is a dimensionless constant. The values of `x` and `y` are respectively :

Options

A. `(1)/(2),(1)/(2)`
B. `-(1)/(2),-(1)/(2)`
C. `(1)/(2),-(1)/(2)`
D. `-(1)/(2),(1)/(2)`

Answer

D

Explanation

`v=Cm^(x)k^(y).`
Writing dimensions on both sides,
`[M^(0)L^(0)T^(-1)]=M^(x)[ML^(0)T^(-2)]^(y)`
`[M^(0)L^(0)T^(-1)]=[M^(x+y)T^(-2y)]`
Comparing dimensions on both sides, we have
`0=x+y`
and `-1=-2yimpliesy=(1)/(2)`
`:.x+(1)/(2)=0impliesx=-(1)/(2)`
So carrect choice is `(d)`.
Aliter. Remembering that frequency of oscillation of loaded spring is `v=(1)/(2pi)sqrt((k)/(m))=(1)/(2pi)(k)^(1//2).m^(-1//2)`
which gives `x=-(1)/(2)andy=(1)/(2)`