Gắn lò xo k vào vật khối lượng ${{m}_{1}}$ được chu kỳ ${{T}_{1}},$
vào vật khối lượng ${{m}_{2}}$ được ${{T}_{2}},$
vào vật khối lượng ${{m}_{1}}+{{m}_{2}}$ được chu kỳ ${{T}_{3}},$
vào vật khối lượng ${{m}_{1}}-{{m}_{2}}\left( {{m}_{1}}>{{m}_{2}} \right)$ được chu kỳ ${{T}_{4}}.$
$\begin{align}& {{T}_{3}}=2\pi \sqrt{\dfrac{{{m}_{1}}+{{m}_{2}}}{k}}\Rightarrow T_{3}^{2}=4{{\pi }^{2}}\dfrac{{{m}_{1}}+{{m}_{2}}}{k} \\ & {{T}_{1}}=2\pi \sqrt{\dfrac{{{m}_{1}}}{k}}\Rightarrow T_{1}^{2}=4{{\pi }^{2}}\dfrac{{{m}_{1}}}{k} \\ & {{T}_{2}}=2\pi \sqrt{\dfrac{{{m}_{2}}}{k}}\Rightarrow T_{2}^{2}=4{{\pi }^{2}}\dfrac{{{m}_{2}}}{k} \\ \end{align}$ $\Rightarrow \left\{ \begin{align}& {{m}_{1}}+{{m}_{2}}=\dfrac{{{T}_{3}}^{2}.k}{4{{\pi }^{2}}} \\ & {{m}_{1}}=\dfrac{T_{1}^{2}.k}{4{{\pi }^{2}}} \\ & {{m}_{2}}=\dfrac{T_{2}^{2}.k}{4{{\pi }^{2}}} \\ \end{align} \right.$ $\Rightarrow \dfrac{{{T}_{3}}^{2}.k}{4{{\pi }^{2}}}=\dfrac{T_{1}^{2}.k}{4{{\pi }^{2}}}+\dfrac{T_{2}^{2}.k}{4{{\pi }^{2}}}\Leftrightarrow T_{3}^{2}=T_{1}^{2}+T_{2}^{2}$
$\begin{align}& {{T}_{4}}=2\pi \sqrt{\dfrac{{{m}_{1}}-{{m}_{2}}}{k}}\Rightarrow T_{4}^{2}=4{{\pi }^{2}}\dfrac{{{m}_{1}}-{{m}_{2}}}{k} \\ & {{T}_{1}}=2\pi \sqrt{\dfrac{{{m}_{1}}}{k}}\Rightarrow T_{1}^{2}=4{{\pi }^{2}}\dfrac{{{m}_{1}}}{k} \\ & {{T}_{2}}=2\pi \sqrt{\dfrac{{{m}_{2}}}{k}}\Rightarrow T_{2}^{2}=4{{\pi }^{2}}\dfrac{{{m}_{2}}}{k} \\ \end{align}$ $\Rightarrow \left\{ \begin{align}& {{m}_{1}}-{{m}_{2}}=\dfrac{{{T}_{4}}^{2}.k}{4{{\pi }^{2}}} \\ & {{m}_{1}}=\dfrac{T_{1}^{2}.k}{4{{\pi }^{2}}} \\ & {{m}_{2}}=\dfrac{T_{2}^{2}.k}{4{{\pi }^{2}}} \\ \end{align} \right.$ $\Rightarrow \dfrac{{{T}_{4}}^{2}.k}{4{{\pi }^{2}}}=\dfrac{T_{1}^{2}.k}{4{{\pi }^{2}}}-\dfrac{T_{2}^{2}.k}{4{{\pi }^{2}}}\Leftrightarrow T_{4}^{2}=T_{1}^{2}-T_{2}^{2}$
Vậy $T_{3}^{2}=T_{1}^{2}+T_{2}^{2}$ và $T_{4}^{2}=T_{1}^{2}-T_{2}^{2}$
$T_{32} = T_{12}+ T_{22} = 52 = 25$
$T_{42} = T_{12}– T_{22} = 32 = 9$
$\Rightarrow T_{32} + T_{42} = 2T_{12}= 34 $ $\Rightarrow T_1 = $$\sqrt{\text{17}}\text{ s}$
$\Rightarrow T_{32} – T_{42} = 2T_{22}= 16 $ $\Rightarrow T_2 =$ $2\sqrt{\text{2}}\text{ s}$