T[m_]:=2 Pi Sqrt[(m+ms/3)/k]

nlm[data_,ms0_,k0_]:=NonlinearModelFit[data,{T[m],k>0,ms≥0},{{ms,ms0},{k,k0}},m,ConfidenceLevel->0.95];

data={{25,12},{50,14},{75,18},{100,19},{125,20},{150,23},{175,24},{200,27},{225,27},{250,29},{275,31}};
ms0=50;
k0=13;
mMax=300;

dataTickner={{10,0.956},{20,1.003},{30,1.037},{40,1.104},{50,1.12},{60,1.157},{70,1.21},{80,1.27},{90,1.32}};
ms0=50;
k0=300;
mMax=100;

dataAlshumemry={{20,0.447},{30,0.542},{40,0.555},{50,0.62},{60,0.663},{70,0.703},{80,0.744},{90,0.79},{100,0.82}};
mMax=100;
k0=1500;

dataVillamizar={{100,0.85},{150,1.008},{200,1.161},{250,1.28},{300,1.393},{350,1.513},{400,1.616},{450,1.695},{500,1.796}};
k0=6200;
mMax=500;

data=dataAlshumemry;

modle=nlm[data,ms0,k0];
Show[ListPlot[data,PlotStyle->Directive[Black,PointSize->0.02]],Plot[modle[m],{m,0,mMax},PlotStyle->Directive[Thick,Dashed,Red]],PlotLabel->"Nonlinear Model", AxesLabel->{"mass (g)","period (s)"},Frame->True]
modle["ParameterConfidenceIntervalTable"]
modle["RSquared"]

dataLinear=Table[{data[[i]][[1]],data[[i]][[2]]^2},{i,1,Length[data]}];
modellinear=LinearModelFit[dataLinear,m,m];
Show[ListPlot[dataLinear,PlotStyle->Directive[Black,PointSize->0.02]],Plot[modellinear[m],{m,0,mMax},PlotStyle->Directive[Thick,Dashed,Red]],Frame->True,AxesLabel->{"mass (g)","period squared (s^2)"},PlotLabel->"Linear Model"]
Normal[modellinear]
modellinear["BestFitParameters"]
modellinear["ParameterConfidenceIntervalTable"]
modellinear["RSquared"]
4 Pi^2/modellinear["BestFitParameters"][[2]]
3 modellinear["BestFitParameters"][[1]]/modellinear["BestFitParameters"][[2]]