土の基本的物理量の変換式

穴に当てはまる数式を脳内計算できたら解答を見る

一部は途中計算も載せてある

体積

$ V_v = eV_s

$ V_t = V_s+V_v=(1+e)V_s

$ V_w = eS_rV_s

$ V_a = e(1-S_r)V_s

重量

$ W_w=wW_s

$ W_t=(1+w)W_s

$ W_w=\frac{w}{1+w}W_t

$ V_s=\frac{1}{1+e}V_t

$ V_v=\frac{e}{1+e}V_t

$ W_s=\frac{1}{1+w}W_t

$ W_w=\frac{w}{1+w}W_t

table:anki

\(V_s=\){{c1::\(\frac{1}{1+e}\)}}\(V_t\)

\(V_v=\){{c1::\(\frac{e}{1+e}\)}}\(V_t\)

\(W_s=\){{c1::\(\frac{1}{1+w}\)}}\(W_t\)

\(W_w=\){{c1::\(\frac{w}{1+w}\)}}\(W_t\)

$ e=\frac{V_v}{V_s}=\frac{V_t-V_s}{V_s}=\frac{\frac{W_s}{V_s}}{\frac{W_s}{V_t}}-1=\frac{\gamma_w}{\gamma_d}G_s-1

解釈

$ e=\frac{\gamma_s-\gamma_d}{\gamma_d}=\frac{{\gamma_d}^{-1}-{\gamma_s}^{-1}}{{\gamma_s}^{-1}}

$ w=\frac{W_w}{W_s}=\frac{W_t-W_s}{W_s}=\frac{\gamma_t-\gamma_d}{\gamma_d}=\frac{\gamma_t}{\gamma_d}-1

$ S_r=\frac{V_w}{V_v}=\frac{1}{eV_s}\frac{W_w}{\gamma_w}=\frac{w}{e}\frac{W_s}{\gamma_wV_s}=\frac{w}{e}G_s

単位体積重量の逆数を使ってこう展開することもできる

$ S_r=\frac{V_w}{V_v}=\frac{{\gamma_w}^{-1}\frac{W_w}{W_s}}{{\gamma_d}^{-1}-{\gamma_s}^{-1}}=\frac{w}{e}\frac{{\gamma_w}^{-1}}{{\gamma_s}^{-1}}=\frac{w}{e}G_s

$ e=wG_s{S_r}^{-1}

table:anki

\(e=\){{c1::\(\frac{\gamma_w}{\gamma_d}G_s-1\)}} \(e=\frac{V_v}{V_s}=\frac{V_t-V_s}{V_s}=\frac{\frac{W_s}{V_s}}{\frac{W_s}{V_t}}-1=\frac{\gamma_w}{\gamma_d}G_s-1\)

\(w=\){{c1::\(\frac{\gamma_t}{\gamma_d}-1\)}} \(w=\frac{W_w}{W_s}=\frac{W_t-W_s}{W_s}=\frac{\gamma_t-\gamma_d}{\gamma_d}=\frac{\gamma_t}{\gamma_d}-1\)

\(S_r=\){{c1::\(\frac{w}{e}G_s\)}} \(S_r=\frac{V_w}{V_v}=\frac{1}{eV_s}\frac{W_w}{\gamma_w}=\frac{w}{e}\frac{W_s}{\gamma_wV_s}=\frac{w}{e}G_s\)

単位体積重量

$ \gamma_s=\frac{W_s}{V_s}=G_s\gamma_w

$ \gamma_s=\frac{W_s}{V_s}=\frac{\frac{1}{1+w}W_t}{\frac{1}{1+e}V_t}=\frac{1+e}{1+w}\gamma_t

$ \gamma_t=\frac{W_t}{V_t}

$ \gamma_d=\frac{W_s}{V_t}

$ \gamma_{sat}=\frac{W_s+\gamma_wV_v}{V_t}

$ \gamma_s=\frac{W_s}{V_s}=\frac{\frac{1}{1+w}W_t}{\frac{1}{1+e}V_t}=\frac{1+e}{1+w}\gamma_t

$ \gamma_d=\frac{W_s}{V_t}=\frac{\frac{1}{1+w}W_t}{V_t}=\frac{1}{1+w}\gamma_t

$ \gamma_{sat}=\frac{W_s+\gamma_wV_v}{V_t}=\gamma_d+\frac{e}{1+e}\gamma_w=\frac{1}{1+w}\gamma_t+\frac{e}{1+e}\frac{1+e}{1+w}\frac{1}{G_s}\gamma_t=\frac{G_s+e}{1+w}\frac{1}{G_s}\gamma_t

これは覚えなくていいや

table:anki

\(\gamma_s=\){{c1::\(\frac{1+e}{1+w}\)}}\(\gamma_t\) \(\gamma_s=\frac{W_s}{V_s}=\frac{\frac{1}{1+w}W_t}{\frac{1}{1+e}V_t}=\frac{1+e}{1+w}\gamma_t\)

\(\gamma_d=\){{c1::\(\frac{1}{1+w}\)}}\(\gamma_t\) \(\gamma_d=\frac{W_s}{V_t}=\frac{\frac{1}{1+w}W_t}{V_t}=\frac{1}{1+w}\gamma_t\)

$ \gamma_s=\frac{1}{\frac{1}{1+e}}\gamma_d=(1+e)\gamma_d

$ \gamma_t=(1+w)\gamma_d

$ \gamma_{sat}=(1+w{S_r}^{-1})\gamma_d=\frac{G_s+e}{G_s}\gamma_d

$ \gamma_s=G_s\gamma_w

$ \gamma_d=\frac{1}{1+e}G_s\gamma_w=\frac{G_s+e\cdot0}{1+e}\gamma_w=\frac{G_s}{1+w{S_r}^{-1}G_s}\gamma_w

$ \gamma_t=\frac{1+w}{1+e}G_s\gamma_w=\frac{G_s+eS_r}{1+e}\gamma_w=\frac{1+w}{1+w{S_r}^{-1}G_s}G_s\gamma_w

$ \gamma_{sat}=\frac{1+w{S_r}^{-1}}{1+e}G_s\gamma_w=\frac{G_s+e\cdot1}{1+e}\gamma_w=\frac{1+w{S_r}^{-1}}{1+w{S_r}^{-1}G_s}G_s\gamma_w

table:anki

\(\gamma_d=\){{c1::\(\frac{G_s+e\cdot0}{1+e}\)}}\(\gamma_w\) \(\gamma_d=\frac{1}{1+e}\gamma_s=\frac{G_s+e\cdot0}{1+e}\gamma_w\)

\(\gamma_t=\){{c1::\(\frac{G_s+eS_r}{1+e}\)}}\(\gamma_w\) \(\gamma_t=\frac{1+w}{1+e}\gamma_s=\frac{G_s+eS_r}{1+e}\gamma_w\)

\(\gamma_{sat}=\){{c1::\(\frac{G_s+e\cdot1}{1+e}\)}}\(\gamma_w=\){{c1::\(\frac{1+w{S_r}^{-1}}{1+w{S_r}^{-1}G_s}G_s\)}}\(\gamma_w\)

$ \gamma_s=\frac{1+e}{1+w{S_r}^{-1}}\gamma_{sat}

$ \gamma_d=\frac{1}{1+w{S_r}^{-1}}\gamma_{sat}

$ \gamma_t=\frac{1+w}{1+w{S_r}^{-1}}\gamma_{sat}

検算