Question

$$T=W\sqrt{\frac{K}{M_{1}+M_{2}}}$$

Solve for K

$\left\{\begin{matrix}K=\left(\frac{T}{W}\right)^{2}\left(M_{1}+M_{2}\right)\text{, }&\left(W<0\text{ or }T\geq 0\right)\text{ and }\left(W>0\text{ or }T\leq 0\right)\text{ and }W\neq 0\text{ and }M_{1}\neq -M_{2}\\K\geq 0\text{, }&T=0\text{ and }W=0\text{ and }M_{1}>-M_{2}\\K\leq 0\text{, }&T=0\text{ and }W=0\text{ and }M_{1}<-M_{2}\\K=0\text{, }&M_{1}\neq -M_{2}\text{ and }T=0\text{ and }W=0\end{matrix}\right.$

Show Solution

Solve for K (complex solution)

$\left\{\begin{matrix}K=\left(\frac{T}{W}\right)^{2}\left(M_{1}+M_{2}\right)\text{, }&W\neq 0\text{ and }M_{1}\neq -M_{2}\text{ and }\left(T=0\text{ or }|arg(\sqrt{\frac{T^{2}}{W^{2}}}W)-arg(T)|<\pi \right)\\K\in \mathrm{C}\text{, }&T=0\text{ and }W=0\text{ and }M_{1}\neq -M_{2}\end{matrix}\right.$

Solve for M_1 (complex solution)

$\left\{\begin{matrix}M_{1}=\left(\frac{W}{T}\right)^{2}K-M_{2}\text{, }&W\neq 0\text{ and }K\neq 0\text{ and }|arg(\sqrt{\left(\frac{T}{W}\right)^{2}}W)-arg(T)|<\pi \text{ and }T\neq 0\\M_{1}\neq -M_{2}\text{, }&\left(W=0\text{ or }K=0\right)\text{ and }T=0\end{matrix}\right.$

Solve for M_1

$\left\{\begin{matrix}M_{1}=\left(\frac{W}{T}\right)^{2}K-M_{2}\text{, }&\left(K\neq 0\text{ and }W<0\text{ and }T<0\right)\text{ or }\left(K\neq 0\text{ and }W>0\text{ and }T>0\right)\\M_{1}\neq -M_{2}\text{, }&T=0\text{ and }K=0\\M_{1}>-M_{2}\text{, }&K\geq 0\text{ and }T=0\text{ and }W=0\\M_{1}<-M_{2}\text{, }&K\leq 0\text{ and }T=0\text{ and }W=0\end{matrix}\right.$