Question

The frequency of vibration of a mass m suspended from a spring constant k us given by the relation F=Cm^xK^y is the dimensionless constant determine x and y

Answer 1

The frequency of vibration of a mass $m$ suspended from a spring constant $k$ us given by the relation

$$f\sim m^xk^y$$

Let's apply the dimensional analysis. We get

$$[f]=T^{-1},\; [m]=M,\; [k]=MT^{-2}$$

$$T^{-1}=M^x(MT^{-2})^y$$

Hence

$$x+y=0,\\-2y=-1.$$

Finally

$$y=1/2,\; x=-1/2$$

$$f\sim\sqrt{\frac{k}{m}}$$