Winch

The steel rope has a diameter of 6mm and a length of 20m.
We are winding on drum width 60mm, starting diameter 50mm. What is the final diameter after winding?

Correct result:

D₂ =  138 mm

Solution:

$$\begin{gathered} D_1=50\text{ mm } \\ {l}_1=60\text{ mm } \\ d=6\text{ mm } \\ {l}_2=20\text{ m}\to \text{ mm}=20\cdot 1000\text{ mm}=20000\text{ mm } \\ {r}_1=D_1\mathrm{/}2=50\mathrm{/}2=25\text{ mm } \\ {V}_1=\pi \cdot r_1^2\cdot l_1=3.1416\cdot 25^2\cdot 60\doteq 117809.7245{\text{mm}}^3 \\ {V}_2=d^2\cdot l_2+V_1=6^2\cdot 20000+117809.7245\doteq 837809.7245{\text{mm}}^3 \\ {V}_2=\pi \cdot r_2^2\cdot l_1 \\ {r}_2=\sqrt{{\displaystyle \frac{V_2}{\pi \cdot l_1}}}=\sqrt{{\displaystyle \frac{837809.7245}{3.1416\cdot 60}}}\doteq 66.6687\text{ mm } \\ {D}_3=2\cdot r_2=2\cdot 66.6687\doteq 133.3374\text{ mm } \\ {D}_2=d\cdot \lceil D_3\mathrm{/}d\rceil =6\cdot \lceil 133.3374\mathrm{/}6\rceil =138\text{ mm} \end{gathered}$$

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