$ V_s =\displaystyle\frac{ \pi }{3} r ^2 h $

V_s = pi * r ^2* h /3

$ M_s = \rho_s V_s $

M_s = rho_s * V_s

$ H = \mu h $

H = mu * h

$ R = \gamma r $

R = gamma * r

$V_f=\displaystyle\frac{\pi}{3}hr^2\mu\gamma^2$

V_f=pi*h*r^2*mu*gamma^2/3

$V_f=\displaystyle\frac{2\pi}{3}hr^2\mu\gamma^2$

V_f=2*pi*h*r^2*mu*gamma^2/3

$M_t=\displaystyle\frac{\pi}{3}hr^2(\rho_s+c_f\mu\gamma^2\rho_f)$

M_t=(pi/3)hr^2(rho_s+c_f*mu*gamma^2*rho_f)

$z_s=\displaystyle\frac{h}{4}$

z_s=h/4

$z_f=(1-\displaystyle\frac{3}{4}\mu)h$

z_f=(1-3*mu/4)*h

$z_f=(1-\displaystyle\frac{1}{2}\mu)h$

z_f=(1-mu/2)*h

$F=Mg\delta\theta$

F=M*g*dtheta

$T=z_{CM}Mg\delta\theta$

T=z_CM*M*g*dtheta

$T=\displaystyle\frac{E\pi r^4}{c_{CM}h}\delta\theta$

T=E pi r^4/(c_CM h) dtheta

$\displaystyle\frac{r^2}{h^3}=\displaystyle\frac{c_{CM}^2\rho_e g}{12E}$

r^2/h^3=c_CM^2 rho_e g/12E

$c_a=\displaystyle\frac{c_{CM}^2\rho_e g}{12 E}$

c_a = c_CM ^2 * rho_e * g /(12* E )

$ r = c_a h ^{3/2}$

r = c_a * h ^(3/2)

$ T_{max} =\displaystyle\frac{2 R ^3}{3} f $

T_max =2* f * R ^3/3

$c_{CM}=\displaystyle\frac{\rho_s+(4-c_c\mu)c_f\mu\gamma^2\rho_f}{\rho_s+c_f\mu\gamma^2\rho_f}$

c_CM=(rho_s+(4-c_c*mu)*c_f*mu*gamma^2*rho_f)/(rho_s+c_f*mu*gamma^2*rho_f)

$z_{CM}=c_{CM}\displaystyle\frac{h}{4}$

z_CM=c_CM*h/4