# V-belt

Calculate a length of the V-belt when the diameter of the pulleys is:

D1 = 600 mm
D2 = 120 mm
d = 480 mm

Result

l =  2185 mm

#### Solution:

$d_{ 1 } = 600 \ mm \ \\ d_{ 2 } = 120 \ mm \ \\ d = 480 \ mm \ \\ \ \\ r_{ 1 } = d_{ 1 }/2 = 600/2 = 300 \ mm \ \\ r_{ 2 } = d_{ 2 }/2 = 120/2 = 60 \ mm \ \\ \ \\ a = r_{ 1 } - r_{ 2 } = 300 - 60 = 240 \ mm \ \\ \ \\ A = \arctan(a/d) = \arctan(240/480) \doteq 0.4636 \ rad \ \\ \ \\ A_{ 1 } = \pi + 2 \cdot \ A = 3.1416 + 2 \cdot \ 0.4636 \doteq 4.0689 \ rad \ \\ A_{ 2 } = \pi - 2 \cdot \ A = 3.1416 - 2 \cdot \ 0.4636 \doteq 2.2143 \ rad \ \\ \ \\ l_{ 1 } = r_{ 1 } \cdot \ A_{ 1 } = 300 \cdot \ 4.0689 \doteq 1220.6664 \ mm \ \\ l_{ 2 } = r_{ 2 } \cdot \ A_{ 2 } = 60 \cdot \ 2.2143 \doteq 132.8578 \ mm \ \\ \ \\ b = \sqrt{ d^2-a^2 } = \sqrt{ 480^2-240^2 } = 240 \ \sqrt{ 3 } \ mm \doteq 415.6922 \ mm \ \\ \ \\ l = l_{ 1 } + l_{ 2 } + 2 \cdot \ b = 1220.6664 + 132.8578 + 2 \cdot \ 415.6922 \doteq 2184.9086 = 2185 \ \text { mm }$

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