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?

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slatington, pa, usa- Joined
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01 May '08 06:46

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*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?**

Solving for r gives 262.5 mm.

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slatington, pa, usa01 May '08 09:58

Thats not what I got, I did the math and then rolled it to confirm.*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.

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01 May '08 10:193 editsPin 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.- Joined
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slatington, pa, usa02 May '08 07:06

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?*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.- Joined
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podunk, PA06 May '08 01:53

I don't know if everyone is rounding, but i get a radius of 337.62*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

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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...- Joined
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slatington, pa, usa06 May '08 11:04

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?*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...- Joined
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06 May '08 11:511 editI 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%