Winch

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

Correct answer:

D₂ =  138 mm

Step-by-step explanation:

$$\begin{gathered} D_1=50\text{mm} \\ {l}_1=60\text{mm} \\ d=6\text{mm} \\ {l}_2=20\text{m}\to \text{mm}=20\cdot 1000\text{mm}=20000\text{mm} \\ {r}_1=D_1/2=50/2=25\text{mm} \\ {V}_1=\pi \cdot r_1^2\cdot l_1=3.1416\cdot 25^2\cdot 60\doteq 117809.7245{\text{mm}}^3 \\ {V}_2=d^2\cdot l_2+V_1=6^2\cdot 20000+\frac{54310283}{461}\doteq 837809.7245{\text{mm}}^3 \\ {V}_2=\pi \cdot r_2^2\cdot l_1 \\ {r}_2=\sqrt{\frac{V_2}{\pi \cdot l_1}}=\sqrt{\frac{837809.7245}{3.1416\cdot 60}}\doteq 66.6687\text{mm} \\ {D}_3=2\cdot r_2=2\cdot 66.6687\doteq 133.3374\text{mm} \\ {D}_2=d\cdot \lceil D_3/d\rceil =6\cdot \lceil 133.3374/6\rceil =138\text{mm} \end{gathered}$$

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