radius of curvature of tapered pin.

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?

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.

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.

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.

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?

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
🙂

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...

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?

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%