Results 1 to 8 of 8

Math Help - circle problem

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    19

    circle problem

    A narrow belt is used to drive a 50 cm diameter pulley from a 10 cm diameter pulley. The centers of the pulleys are 50 cm apart.
    a.How long must the belt be if the pulleys are rotating in the same direction?
    b How long must the belt be if the pulleys are rotating in the opposite direction...(the belt crosses over itself between the two pulleys)?
    a. one pulley is 50 cm so its radius is 25cm
    the other pulley is 10 cm so its radius is 5 cm
    they are 50 cm apart from their centers.
    50+(25+5)=80
    the belt needs to be 80 cm long. Is that correct?
    I'm not sure how to do part b of this question.

    In addition, does anyone know where I can read more about situations like this so I can understand the situation better?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,807
    Thanks
    116
    Quote Originally Posted by yoleven View Post
    A narrow belt is used to drive a 50 cm diameter pulley from a 10 cm diameter pulley. The centers of the pulleys are 50 cm apart.
    a.How long must the belt be if the pulleys are rotating in the same direction?
    b How long must the belt be if the pulleys are rotating in the opposite direction...(the belt crosses over itself between the two pulleys)?
    a. one pulley is 50 cm so its radius is 25cm
    the other pulley is 10 cm so its radius is 5 cm
    they are 50 cm apart from their centers.
    50+(25+5)=80
    the belt needs to be 80 cm long. Is that correct? No
    I'm not sure how to do part b of this question.

    In addition, does anyone know where I can read more about situations like this so I can understand the situation better?
    Unfortunately I haven't much time yet so I only can give you the sketches to questions a) and b).

    The belt is drawn in red. The triangles marked in red are right triangles whose dimensions are given so you are able to calculate distances (by Pythagorean theorem) and angles (by Cosine) which you need to calculate the arcs on the pulleys.

    r_s = small\ radius
    r_l = large\ radius
    Attached Thumbnails Attached Thumbnails circle problem-scheiben_selberichtung.png   circle problem-scheiben_entgegenrichtung.png  
    Last edited by earboth; September 30th 2008 at 12:38 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,807
    Thanks
    116
    Quote Originally Posted by yoleven View Post
    A narrow belt is used to drive a 50 cm diameter pulley from a 10 cm diameter pulley. The centers of the pulleys are 50 cm apart.
    a.How long must the belt be if the pulleys are rotating in the same direction?
    ...
    Here I am again:

    I'm referring to my sketch #1 in my previous post.

    1. The length of the sides of the right triangle are:

    80 cm: distance between the midpoints of the pulleys
    40 cm : difference of the 2 radii
    therefore s = \sqrt{80^2-40^2} = 40\cdot\sqrt{3}

    2. The interior angle at the midpoint of the large pulley is \phi:

    \cos(\phi) = \frac{40}{80}=\frac12~\implies~\phi=60^\circ

    Therefore the central angle of the arc a_l is \alpha= 240^\circ (Why?)
    and the central angle of the arc a_s is \beta = 120^\circ

    3. The length of the belt is calculated by:

    L_{belt} = \underbrace{2\cdot 40\cdot\sqrt{3}}_{straight\ lines}+\underbrace{\frac{240}{360}\cdot 2\pi\cdot 50}_{large\ arc} + \underbrace{\frac{120}{360}\cdot 2\pi\cdot 10}_{small\ arc} = 80\cdot\sqrt{3} + \frac{220}3 \pi \approx 368.95\ cm

    to #b) Use my sketch to do this problem:
    1. Calculate the straight line s
    2. Calculate the angle \alpha
    3. Calculate the length of the arcs

    I've got as total length of the belt: 396.09 cm
    Last edited by earboth; September 30th 2008 at 01:24 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Sep 2008
    Posts
    19
    Wow. Thanks. Those are really good sketches.
    I hadn't expected that.
    I have a few questions but the first one is, when you are figuring out the side of the right triangle in sketch #1, How do you know to take 40cm, (the difference of the two radi)?
    How do you know that"s what you are going to do?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,807
    Thanks
    116
    Quote Originally Posted by yoleven View Post
    Wow. Thanks. Those are really good sketches.
    I hadn't expected that.
    I have a few questions but the first one is, when you are figuring out the side of the right triangle in sketch #1, How do you know to take 40cm, (the difference of the two radi)?
    How do you know that"s what you are going to do?
    1. The straight parts of the belt must be tangent to both circles. The tangent in question is translated so that it passes through the midpoint of the smaller circle and the larger circle is "shrunken"(?) by the length of the radius of the smaller circle.
    If you look at those circles drawn by a dotted line then you have the this situation.

    2. A tangent to a circle is perpendicular to the radius at the tangent point. I used this property to construct the right triangles which you need to calculate all missing values.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Sep 2008
    Posts
    19
    Okay, that makes sense to me. I can see how you got an angle of 240 in the larger wheel but I cant see how you got 120 in the smaller wheel. Could you explain this?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,807
    Thanks
    116
    Quote Originally Posted by yoleven View Post
    Okay, that makes sense to me. I can see how you got an angle of 240 in the larger wheel but I cant see how you got 120 in the smaller wheel. Could you explain this?
    The radii of the 2 circles must be parallel because they are perpendicular to the same tangent. That means you have 2 parallels which are crossed by a straight line which connects the 2 midpoints. Now examine the angles at the midpoints: alternate angles, corresponding angles, etc.

    As a result you'll get that the 2 yellow marked angles add up to 360. So if one angle is 240 the second one must be 120.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Newbie
    Joined
    Sep 2008
    Posts
    19
    Okay, I can see that. Thanks a lot for all your help!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Circle problem
    Posted in the Advanced Math Topics Forum
    Replies: 3
    Last Post: May 20th 2011, 10:20 AM
  2. Circle Problem
    Posted in the Geometry Forum
    Replies: 2
    Last Post: November 8th 2010, 03:40 PM
  3. circle problem
    Posted in the Geometry Forum
    Replies: 8
    Last Post: April 23rd 2010, 10:13 AM
  4. Circle Problem
    Posted in the Geometry Forum
    Replies: 5
    Last Post: May 12th 2009, 10:27 AM
  5. circle problem
    Posted in the Geometry Forum
    Replies: 4
    Last Post: November 9th 2008, 11:32 AM

Search Tags


/mathhelpforum @mathhelpforum