# 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?

Result

D2 =  138 mm

#### Solution:

$D_{ 1 } = 50 \ mm \ \\ l_{ 1 } = 60 \ mm \ \\ d = 6 \ mm \ \\ l_{ 2 } = 20 \ m = 20 \cdot \ 1000 \ mm = 20000 \ mm \ \\ \ \\ r_{ 1 } = D_{ 1 }/2 = 50/2 = 25 \ mm \ \\ \ \\ V_{ 1 } = \pi \cdot \ r_{ 1 }^2 \cdot \ l_{ 1 } = 3.1416 \cdot \ 25^2 \cdot \ 60 \doteq 117809.7245 \ mm^3 \ \\ V_{ 2 } = d^2 \cdot \ l_{ 2 } + V_{ 1 } = 6^2 \cdot \ 20000 + 117809.7245 \doteq 837809.7245 \ mm^3 \ \\ \ \\ V_{ 2 } = \pi \cdot \ r_{ 2 }^2 \cdot \ l_{ 1 } \ \\ \ \\ r_{ 2 } = \sqrt{ \dfrac{ V_{ 2 } }{ \pi \cdot \ l_{ 1 } } } = \sqrt{ \dfrac{ 837809.7245 }{ 3.1416 \cdot \ 60 } } \doteq 66.6687 \ mm \ \\ \ \\ D_{ 3 } = 2 \cdot \ r_{ 2 } = 2 \cdot \ 66.6687 \doteq 133.3374 \ mm \ \\ D_{ 2 } = d \cdot \ \lceil D_{ 3 }/d \rceil = 6 \cdot \ \lceil 133.3374/6 \rceil = 138 = 138 \ \text { mm }$

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#### Following knowledge from mathematics are needed to solve this word math problem:

Do you want to convert length units? Do you know the volume and unit volume, and want to convert volume units?

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