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Math Help - Radian question

  1. #1
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    Radian question

    A flat belt is wrapped around two pulleys of diameter 10cm and 16cm seperated by a distance of 26cm as shown below.
    Find the exact length of the belt.

    Thanks guys.

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  2. #2
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    Quote Originally Posted by deltaxray View Post
    A flat belt is wrapped around two pulleys of diameter 10cm and 16cm seperated by a distance of 26cm as shown below.
    Find the exact length of the belt.

    Thanks guys.

    I'm not sure what you meant by "distance of 26 cm":
    - Is this the distance between the centres?
    - Is this the clear space between the pulleys?

    I've attached a correct construction of the inner common tangent of 2 circles.

    You are dealing with 2 similar right triangles. The ratio of distances of these triangles depends of the ratio of the two radii.
    Attached Thumbnails Attached Thumbnails Radian question-commtang2circles.png  
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  3. #3
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    Quote Originally Posted by deltaxray View Post
    A flat belt is wrapped around two pulleys of diameter 10cm and 16cm seperated by a distance of 26cm as shown below.
    Find the exact length of the belt.

    Thanks guys.

    Let r be the radius of the smaller pulley.
    Let R be the radius of the larger pulley.
    Let s be the smallest distance between the edges of the pulleys (r+s+R = distance be pulley centers).
    Let \theta be the acute angle where the belts 'intersect' -- theta in RADIANS.

    \theta = 2 \cdot arcsin \left( \dfrac{R+r}{R+s+r}\right)

    PerimeterLength =  (r+R)(\pi+\theta) + 2\sqrt{(R+s+r)^2 - (R+r)^2}

    Substitute your specific values into the equation.

    .
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