Posers and Puzzles

Posers and Puzzles

  1. Standard membersonhouse
    Fast and Curious
    slatington, pa, usa
    Joined
    28 Dec '04
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    52874
    01 May '08 05:12
    I am holding in my hand a very precisely made taper pin, its 75.000 mm long, 18.000 mm wide at the top and 14.000 mm wide at the bottom. If you roll it on a table it will describe a circle. What is the diameter of that circle?
  2. Joined
    13 Dec '06
    Moves
    792
    01 May '08 06:46
    Originally posted by sonhouse
    I am holding in my hand a very precisely made taper pin, its 75.000 mm long, 18.000 mm wide at the top and 14.000 mm wide at the bottom. If you roll it on a table it will describe a circle. What is the diameter of that circle?
    Suppose the bottom of the pin describes a circle with radius r (mm). Then the circumference of that circle is 2.pi.r, so the pin undergoes 2.pi.r/(14.pi)=r/7 revolutions when it describes this circle. Then the top of the pin describes a circle of radius r+75 and circumference (r/7)*(18.pi). So we have 2.pi*(r+75) = 18/7.pi.r

    Solving for r gives 262.5 mm.

    I should go to bed.
  3. Standard membersonhouse
    Fast and Curious
    slatington, pa, usa
    Joined
    28 Dec '04
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    52874
    01 May '08 09:58
    Originally posted by GregM
    Suppose the bottom of the pin describes a circle with radius r (mm). Then the circumference of that circle is 2.pi.r, so the pin undergoes 2.pi.r/(14.pi)=r/7 revolutions when it describes this circle. Then the top of the pin describes a circle of radius r+75 and circumference (r/7)*(18.pi). So we have 2.pi*(r+75) = 18/7.pi.r

    Solving for r gives 262.5 mm.

    I should go to bed.
    Thats not what I got, I did the math and then rolled it to confirm.
  4. Joined
    07 Sep '05
    Moves
    35068
    01 May '08 10:193 edits
    Pin has radius r1 at bottom, r2 at top, and length L
    When rolled, describes a circle radius R1 (inner) and R2 (outer)
    This takes n complete rolls

    2.pi.R1 = n.2.pi.r1 => R1 = n.r1
    Similarly R2 = n.r2

    We also know R2 = R1 + L
    => n.r2 = n.r1 + L
    => n = L/(r2 - r1)

    => R1 = L.r1/(r2 - r1), R2 = L.r2/(r2 - r1)

    (Sanity check: if r1 = r2 we get an infinite radius - this is expected as it will roll in a straight line)

    Did you want the diameter of the inner or outer circle?

    Inner = 2.L.r1/(r2 - r1) = 525.0mm
    Outer = 2.L.r2/(r2 - r1) = 675.0mm

    EDIT: This seems to agree with what Greg got.
  5. Standard membersonhouse
    Fast and Curious
    slatington, pa, usa
    Joined
    28 Dec '04
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    52874
    02 May '08 07:06
    Originally posted by mtthw
    Pin has radius r1 at bottom, r2 at top, and length L
    When rolled, describes a circle radius R1 (inner) and R2 (outer)
    This takes n complete rolls

    2.pi.R1 = n.2.pi.r1 => R1 = n.r1
    Similarly R2 = n.r2

    We also know R2 = R1 + L
    => n.r2 = n.r1 + L
    => n = L/(r2 - r1)

    => R1 = L.r1/(r2 - r1), R2 = L.r2/(r2 - r1)

    (Sanity check: if r1 = r2 we get an inf ...[text shortened]... ) = 525.0mm
    Outer = 2.L.r2/(r2 - r1) = 675.0mm

    EDIT: This seems to agree with what Greg got.
    I see now. He calculated the inner circle. I was thinking of the outer circle and got 675 for that, which is why I said I got something differant. I should have seen there were two circles involved, didn't specify well enough the problem. But I solved it without using PI. Can you figure out how to do it (outer circle) without involving PI?
  6. Backside of desert
    Joined
    09 Nov '06
    Moves
    14828
    05 May '08 16:06
    pin is a truncated cone.

    outside is A=18, inside B=14, change C=4, length Q=75

    C/Q = A/R
    4/75 = 18/R
    R = 337.5
    D = 2R = 675
    🙂
  7. Subscriberjoe shmo
    Strange Egg
    podunk, PA
    Joined
    10 Dec '06
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    7733
    06 May '08 01:53
    Originally posted by preachingforjesus
    pin is a truncated cone.

    outside is A=18, inside B=14, change C=4, length Q=75

    C/Q = A/R
    4/75 = 18/R
    R = 337.5
    D = 2R = 675
    🙂
    I don't know if everyone is rounding, but i get a radius of 337.62

    The way i see it every one is using the hieght of the cone, but the radius is actually the hypotenuse......

    i realize in this situation the difference is negligable, but in another aplication this could make a difference...
  8. Standard membersonhouse
    Fast and Curious
    slatington, pa, usa
    Joined
    28 Dec '04
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    52874
    06 May '08 11:04
    Originally posted by joe shmo
    I don't know if everyone is rounding, but i get a radius of 337.62

    The way i see it every one is using the hieght of the cone, but the radius is actually the hypotenuse......

    i realize in this situation the difference is negligable, but in another aplication this could make a difference...
    It sounds like you are making the circle a bit bigger than the cone, I envision the circle radius points to be touching at each edge of the wider portion of the pin, while you seem to be extending the length of the pin a bit to make the radius 0.12 units longer, which would put the radius in the center of the pin but the pin would have to be longer to make the radius = the legnth of the pin. The way we did it, the radius is = to the edges of the pin. Does that make sense?
  9. Joined
    07 Sep '05
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    35068
    06 May '08 11:511 edit
    I think he's right. The answers we've given assume 75mm measured along the edge, whereas the "length" would usually be measured from the centre of one end to the centre of the other.

    If that case, the length of the side would be 75.027mm [sqrt(75^2 + 2^2)], so you'd need to increase the answers by 0.036%
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