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Solve for U: h = 2 extpi extsqrt((l-U)/(g-U))

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Answer

U=h2g−4π2lh2−4π2U = \frac{h^2g - 4\pi^2l}{h^2 - 4\pi^2}

Step-by-step solution

8 steps

  1. Divide both sides by 2π2\pi.

    h=2πl−Ug−Uh = 2\pi \sqrt{\frac{l-U}{g-U}} h2π=l−Ug−U\frac{h}{2\pi} = \sqrt{\frac{l-U}{g-U}}

  2. Square both sides of the equation to remove the square root.

    (h2π)2=(l−Ug−U)2\left(\frac{h}{2\pi}\right)^2 = \left(\sqrt{\frac{l-U}{g-U}}\right)^2 h24π2=l−Ug−U\frac{h^2}{4\pi^2} = \frac{l-U}{g-U}

  3. Multiply both sides by (g−U)(g-U) to eliminate the denominator.

    h24π2(g−U)=l−U\frac{h^2}{4\pi^2}(g-U) = l-U

  4. Expand the left side of the equation.

    h2g4π2−h2U4π2=l−U\frac{h^2g}{4\pi^2} - \frac{h^2U}{4\pi^2} = l-U

  5. Collect all terms containing UU on one side and terms not containing UU on the other side.

    Add h2U4π2\frac{h^2U}{4\pi^2} to both sides and subtract ll from both sides.

    h2g4π2−l=h2U4π2−U\frac{h^2g}{4\pi^2} - l = \frac{h^2U}{4\pi^2} - U

  6. Factor out UU from the terms on the right side.

    h2g4π2−l=U(h24π2−1)\frac{h^2g}{4\pi^2} - l = U\left(\frac{h^2}{4\pi^2} - 1\right)

  7. Divide both sides by (h24π2−1)\left(\frac{h^2}{4\pi^2} - 1\right) to isolate UU.

    U=h2g4π2−lh24π2−1U = \frac{\frac{h^2g}{4\pi^2} - l}{\frac{h^2}{4\pi^2} - 1}

  8. Simplify the expression by multiplying the numerator and the denominator by 4π24\pi^2.

    U=4π2(h2g4π2−l)4π2(h24π2−1)U = \frac{4\pi^2\left(\frac{h^2g}{4\pi^2} - l\right)}{4\pi^2\left(\frac{h^2}{4\pi^2} - 1\right)} U=h2g−4π2lh2−4π2U = \frac{h^2g - 4\pi^2l}{h^2 - 4\pi^2}
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