Solve F=mg+mv^2/R

$$F = m g + \frac { m v ^ { 2 } } { R }$$

Solve for R

$$\left\{\begin{matrix}R=-\frac{mv^{2}}{gm-F}\text{, }&v\neq 0\text{ and }m\neq 0\text{ and }F\neq gm\\R\neq 0\text{, }&\left(F=gm\text{ and }v=0\right)\text{ or }\left(v\neq 0\text{ and }F=0\text{ and }m=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable R cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by R.

$$FR=mgR+mv^{2}$$

Subtract mgR from both sides.

$$FR-mgR=mv^{2}$$

Reorder the terms.

$$-Rgm+FR=mv^{2}$$

Combine all terms containing R.

$$\left(-gm+F\right)R=mv^{2}$$

The equation is in standard form.

$$\left(F-gm\right)R=mv^{2}$$

Divide both sides by F-mg.

$$\frac{\left(F-gm\right)R}{F-gm}=\frac{mv^{2}}{F-gm}$$

Dividing by F-mg undoes the multiplication by F-mg.

$$R=\frac{mv^{2}}{F-gm}$$

Variable R cannot be equal to 0.

$$R=\frac{mv^{2}}{F-gm}\text{, }R\neq 0$$

Solve for F

$$F=\frac{m\left(v^{2}+Rg\right)}{R}$$

$$R\neq 0$$