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Can anyone solve this problem?

A maths problem I can't solve

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Gary Wooding27/03/2019 10:52:25
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The diagrams show a V-groove in a solid block. The angle of the groove is V.

There is a right circular cone, included angle C, resting in the groove. The point of the cone is at the bottom of the groove, and the cone just lies there touching both sides of the groove. The axis of the cone now lies at some angle X relative to the bottom of the groove.

The problem is simple: what is angle X?

cone in groove.jpgcone in groove2.jpg

Gary Wooding27/03/2019 11:09:10
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Hmm, I've somehow managed add this posting twice. Could a moderator remove one of them please?

Trevorh27/03/2019 11:14:17
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Isn't it just a right angle triangle - the right angle being the base of the V block, the hypotenuse being the centre line X extended to the edge of the V block gives you the Right angle

called a scalene right angled triangle with no equal sides

so the inclusive angle will be 35 degree at the point and 55 degrees at the vertical end

 

regards Trevor

Edited By Trevorh on 27/03/2019 11:16:52

JasonB27/03/2019 11:42:21
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Trevor how do you get those figures when none of the angles are known? or did you just measure off the drawing.

I would have thought Gary was after a formula so that if V and C change you can work out X

Gary Wooding27/03/2019 11:49:16
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Jason is right, I'm looking for a formula that calculates X for various values of C and V.

Paul Lousick27/03/2019 12:09:16
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Easy enough to work out X in a 3D model but need value of C & V

JasonB27/03/2019 12:35:41
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For those that don't have cad and want to see if they can come up with a formula then these are some random CAD values.

V = 80deg

C = 40deg included angle

X = 32.15deg

Andrew Johnston27/03/2019 13:04:29
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I don't have time at the moment to do the analysis. But by inspection we can say that the following inequality is a necessary condition:

C ≤ V

In the limit when C = V then X = 90°.

Andrew

Frances IoM27/03/2019 13:12:50
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I'm sure I've seen this solved - look up analysis of conic sections - was in all advanced geometry books many years ago (eg late Victorian text books) - had a quick look at my bookshelves but can't find the book I was thinking of
JasonB27/03/2019 14:02:21
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This looks like it may do, going to try it with those figures

 

Edited By JasonB on 27/03/2019 14:03:57

JasonB27/03/2019 14:41:17
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yes that works

Sin X = Sin M x Cosec N

Where M is half the value of C and N is half the Value of V from the original question.

So taking those CAD figures I posted earlier

Sin X = Sin 20 x Cosec 40

Sin X = 0.342 x 1.558

Sin X = 0.533

X = 32.21 which allowing for me rounding to 3 decimal places is right.

 

I did have to go and look up what Cosec was as I only went as far as O level mathswink

Edited By JasonB on 27/03/2019 14:42:53

Gary Wooding27/03/2019 14:46:01
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Posted by JasonB on 27/03/2019 14:02:21:

This looks like it may do, going to try it with those figures

Edited By JasonB on 27/03/2019 14:03:57

Thanks Jason, I can verify that formula does work.

How did you do it in CAD? I could only do it by trial and error.

John Haine27/03/2019 14:53:47
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Or sin(x) = sin(M/2)/sin(V/2) since csc(x) = 1/sin(x)

Now to prove it!

I'm never quite sure why sec and cosec were ever invented since they are just reciprocals of cos and sin respectively.

JasonB27/03/2019 14:54:10
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I drew out the vee block and cone to those angles which is easy enough. I then assembled the two using a tangental mate to first mate the slope of the cone to one side of the vee and then a second similar mate to get it to sit against that. Then just a case of measuring the angle between base and the central axis is the cone. I've made the block semi transparent so you can see things better

 

cone angle.jpg

Edited By JasonB on 27/03/2019 14:54:52

SillyOldDuffer27/03/2019 16:02:24
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Posted by John Haine on 27/03/2019 14:53:47:

...

I'm never quite sure why sec and cosec were ever invented since they are just reciprocals of cos and sin respectively.

Me neither, except reading an old book about navigation suggests an answer: it does sums using haversines, covercosines and other weirdness. The reason seems to be simplification and error reduction of calculations at a time when they were done manually in mid-Atlantic by sea-sick navigators in rough weather. It's quicker and safer to look up cosec x in a table than it is to crunch 1 / sin x, or to look up sin x and then it's reciprocal in two tables. The formula involved are simplified too, for example the Haversine formula is hot at calculating great circle distances. Likely the functions also helped on land, perhaps laying out curves on railway lines.

Other examples of 'labour saving' trig functions include Versine, Coversine, Vercosine, Secant, Cosecant, Exsecant, Excosecant, Cotangent and probably others. Once you have a computer, calculator or even a slide rule, the need for them largely disappears, I guess. I've never come across them in real life!

Dave

Gary Wooding27/03/2019 16:13:09
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Thanks Jason,

I tried, and failed to do it in Fusion, which is the reason I wanted a formula. Fusion doesn't use mates - it uses something called joints instead, which I couldn't make work.

John Reese27/03/2019 18:43:59
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Posted by John Haine on 27/03/2019 14:53:47:

I'm never quite sure why sec and cosec were ever invented since they are just reciprocals of cos and sin respectively.

In the days of solving math using pencil & paper it was simpler to multiply by the inverse function than to divide by the function. Some of the early mechanical calculators were not capable of division hence the need for the inverse functions.

Neil Wyatt27/03/2019 20:50:18
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Posted by SillyOldDuffer on 27/03/2019 16:02:24:

Other examples of 'labour saving' trig functions include Versine, Coversine, Vercosine, Secant, Cosecant, Exsecant, Excosecant, Cotangent and probably others. Once you have a computer, calculator or even a slide rule, the need for them largely disappears, I guess. I've never come across them in real life!

Dave

Most of those sound like decongestants.

My brain needs one after reading this thread...

Neil

Chris Trice27/03/2019 23:24:09
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The thing that threw me was that the round disk of the wide part of the cone, when angled forward starts making contact with the V block nearer and nearer to the east and west positions. Looking from the front, the disk starts to become elliptical the more it leans.

Brian Wood28/03/2019 09:53:21
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Jason,

May I ask which book it was you found that explanation in please. My various reference books, including the full set of three Chapman's on Workshop Technology doesn't have it

Regards

Brian

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