Initial program 14.0
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
Initial simplification13.7
\[\leadsto \sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*} \cdot w0\]
- Using strategy
rm Applied div-inv13.7
\[\leadsto \sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{2}}{\color{blue}{d \cdot \frac{1}{D}}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*} \cdot w0\]
Applied associate-/r*14.2
\[\leadsto \sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \color{blue}{\frac{\frac{\frac{M}{2}}{d}}{\frac{1}{D}}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*} \cdot w0\]
- Using strategy
rm Applied associate-/r/13.8
\[\leadsto \sqrt{(\left(\color{blue}{\left(\frac{\frac{M}{2}}{d} \cdot D\right)} \cdot \frac{\frac{\frac{M}{2}}{d}}{\frac{1}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*} \cdot w0\]
- Using strategy
rm Applied add-sqr-sqrt13.8
\[\leadsto \sqrt{\color{blue}{\sqrt{(\left(\left(\frac{\frac{M}{2}}{d} \cdot D\right) \cdot \frac{\frac{\frac{M}{2}}{d}}{\frac{1}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*} \cdot \sqrt{(\left(\left(\frac{\frac{M}{2}}{d} \cdot D\right) \cdot \frac{\frac{\frac{M}{2}}{d}}{\frac{1}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}}} \cdot w0\]
Applied rem-sqrt-square13.8
\[\leadsto \color{blue}{\left|\sqrt{(\left(\left(\frac{\frac{M}{2}}{d} \cdot D\right) \cdot \frac{\frac{\frac{M}{2}}{d}}{\frac{1}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}\right|} \cdot w0\]
Simplified12.1
\[\leadsto \left|\color{blue}{\sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(\left(-\frac{h}{\ell}\right) \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) + 1)_*}}\right| \cdot w0\]
- Using strategy
rm Applied div-inv12.1
\[\leadsto \left|\sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(\left(-\color{blue}{h \cdot \frac{1}{\ell}}\right) \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) + 1)_*}\right| \cdot w0\]
Applied distribute-lft-neg-in12.1
\[\leadsto \left|\sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(\color{blue}{\left(\left(-h\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) + 1)_*}\right| \cdot w0\]
Applied associate-*l*8.2
\[\leadsto \left|\sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \color{blue}{\left(\left(-h\right) \cdot \left(\frac{1}{\ell} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right)\right)} + 1)_*}\right| \cdot w0\]
Final simplification8.2
\[\leadsto \left|\sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{-1}{\ell}\right) \cdot h\right) + 1)_*}\right| \cdot w0\]
