# have x(b) to find b(x)...

Can you rewrite:

x=x(b)= h*tan(b)-(r/cos(b))+r

as b(x)=...

r and h are constants.

Thank you.

Math / Science

Last Comment
Mike Eghtebas

8/22/2022 - Mon
Peter Kwan

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rrz

Look at the bottom of the discussion at
https://www.experts-exchange.com/Programming/Languages/Scripting/Q_27237703.html
The equation should have been
x = h*tan(b) + r/cos(b) - r

So once we get pkwan's corrected equation. We can plug it into our function for "a" and then we will have the long sought after  "a" as a function of "x"  . Wow! It is going to be a complicated equation.
Mike Eghtebas

:rrz@871311,

I guess based on the information given to pkwan, he has done his magic and the answer should be accepted. I guess I will post a new question asking for his help after we first discuss a point.

Looking at the attached graph, I guess you are accepting (eq. 1) and possibly you are rejecting (eq. 2). What is wrong with (eq. 2)? I find it to be correct. If you have a simpler one to replace it, it will be nice to show what it is and how you derive it?

Thank you,

Mike
drum-1.png
Mike Eghtebas

a =a(x,b)= (1/r)*sqrt((h + rsin(b))^2 + (x + r - rcos(b))^2) - b-h/r         (eq.1)
x=x(b)=r (cos(b)+ tan(b)sin(b)-1)+ tan(b)h                                           (eq. 2)
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SOLUTION
rrz

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rrz

If you make a new graph, then change the way you illustrate the rope. It would more clear to have the orange segment followed by the purple segment followed by the red segment. Even better, just show the r*(a+b) rope segment in one color followed by the original rope segment in another color.  As the pulling progresses the drum rotates and the point of contact moves up simultaneously, so the segment of rope that comes off the drum is made from both. I look at it as some unwinding and some unwrapping.
Mike Eghtebas

As I am going through your post one item at a time:

r:> a = (r *tan(b) + h/cos(b) - h)/r - b

I can see how you got this and it is brilliant.
---------------
re:> ((x+d)/rTan(b)) = h/r

I think this should be

((x+d)/rTan(b)) = (h+k)/r

You have switched from one version (left below) of blue triangle to the other one (right below). See the graph below for k and d.

Unfortunately, my last attempt in deriving x =h*Tan(b) - (r/cos(b)) +r had the same problem.

I will read the rest of the post and also wait for your response for the second item here.

Regards,

Mike

drum-3.png
Mike Eghtebas

re: It would more clear to have t...

I just saw your post on this. Yes it is good idea and I will show r(a+b) together at the top in same color.

Mike
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rrz

Forget about d and k.  Just use my derivation that I gave using the horizontal additions.
Mike Eghtebas

re:> x + r = r/cos(b) + h*tan(b)

Wow, this is real nice.
Mike Eghtebas

ok, i will make the graph and the rest.

thx
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Mike Eghtebas

again with x + r = r/cos(b) + h*tan(b)

This x has a moving origin. This cannot be. the origin of x has to be r distance to the right from the center line of the drum.

Sorry for this set back.

Mike
rrz

>the origin of x has to be r distance to the right from the center line of the drum
Yes.
>This x has a moving origin.
No.  You said that "r" is a constant for each animation.
Mike Eghtebas

on the graph above x shown with two dashes has moving origin as the string is pulled.
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rrz

x = 0 at the point where the rope started before being pulled. You have x mislabeled  in the right-hand graph above here.
Mike Eghtebas

the following equation is good one because it doesn't refer to x:

a = (r *tan(b) + h/cos(b) - h)/r - b

We need to fix string origin to zero where it is pulled zero inch.
Mike Eghtebas

brb
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rrz

In the left-hand graph, you have it correct.
x=0 where the rope starts.
The drum is located at  x = - r   and  y = h
rrz

>We need to fix string origin to zero where it is pulled zero inch.
It has been there all along.
>brb
What does that mean?
rrz

rrz>You have x mislabeled  in the right-hand graph above here.
That where you got confused. The bottom of that triangle should have labeled  h*tan(b)
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Mike Eghtebas

rrz@871311,

I agree and very much like  a = (r *tan(b) + h/cos(b) - h)/r - b

But, I don't see how you get this one:

x + r = r/cos(b) + h*tan(b)

Could you please, in reference to the graph below, make your case? I know you have patiently done this a few times but I thing there is something illusive causes the problem going back and fort from one graph to another.

thanks.
Cam-3.png
Mike Eghtebas

wrong graph... sorry

rrz@871311,

I agree and very much like  a = (r *tan(b) + h/cos(b) - h)/r - b

But, I don't see how you get this one:

x + r = r/cos(b) + h*tan(b)

Could you please, in reference to the graph below, make your case? I know you have patiently done this a few times but I thing there is something illusive causes the problem going back and fort from one graph to another.

thanks.
drum-3.png
rrz

The drum is located at x = - r   y = h
The rope starts at x = 0  y = 0
After some pulling  the bottom the rope is at x = x
At that time the horizontal distance between the center of the drum and the bottom of the rope = x + r
https://www.experts-exchange.com/Programming/Languages/Scripting/Q_27237703.html?cid=748#a36323336
The bottom of the yellow triangle = r/cos(b)
The bottom of the blue triangle = h*tan(b)
They cover the same horizontal distance as the radius of the drum and the distance pulled.
Therefore
x + r = r/cos(b) + h*tan(b)

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Mike Eghtebas

This is thinking out of the box. I will post the new question now.

Thanks,

Mike
Mike Eghtebas