System of equations for hanging mobile
$begingroup$
I've been interested for some time in hanging mobiles. (See Alexander Calder, for example)
To simplify, one can start by ignoring the mass of the rods and string, and only consider:
arm lengths
leaf masses
Doing this, we can define a mathematical 'mobile' M as an L-system:
$M -> M_l A_l A_r M_r$
$M -> L$
$A -> (arm) length$
$L -> (leaf) mass$
$l = left$
$r = right$
Where $M_l * A_l = A_r * M_r$ at every level.
This can lead to an interesting system of equations, since a mobile must always balance at every level within each subtree.
E.g.:
$M_{ll} * A_{ll} = A_{lr} * M_{lr}$,
where $ll$ means the left branch of the left branch, starting from the top, and $lr$ means the right branch of the left branch, etc.
But also there are inter-level dependencies of mass, since:
$M_l = M_{ll} + M_{lr}$
I am interested to understand how to model this system of equations and solve it. I see that it can be done with linear algebra, but don't quite know the next step, since the mass dependencies make things more complicated.
Also just ways to think about this system would be helpful.
This is for my own interest, fun, and learning, glad to learn with others.
Thanks for any help!
linear-algebra
asked Dec 29 '18 at 12:44
user7400474user7400474
11
$endgroup$
add a comment |
$begingroup$
I've been interested for some time in hanging mobiles. (See Alexander Calder, for example)
To simplify, one can start by ignoring the mass of the rods and string, and only consider:
arm lengths
leaf masses
Doing this, we can define a mathematical 'mobile' M as an L-system:
$M -> M_l A_l A_r M_r$
$M -> L$
$A -> (arm) length$
$L -> (leaf) mass$
$l = left$
$r = right$
Where $M_l * A_l = A_r * M_r$ at every level.
This can lead to an interesting system of equations, since a mobile must always balance at every level within each subtree.
E.g.:
$M_{ll} * A_{ll} = A_{lr} * M_{lr}$,
where $ll$ means the left branch of the left branch, starting from the top, and $lr$ means the right branch of the left branch, etc.
But also there are inter-level dependencies of mass, since:
$M_l = M_{ll} + M_{lr}$
I am interested to understand how to model this system of equations and solve it. I see that it can be done with linear algebra, but don't quite know the next step, since the mass dependencies make things more complicated.
Also just ways to think about this system would be helpful.
This is for my own interest, fun, and learning, glad to learn with others.
Thanks for any help!
linear-algebra
asked Dec 29 '18 at 12:44
user7400474user7400474
11
$endgroup$
$begingroup$
Possibly related, A numberphile video in the context of balancing centrifuges and when this can be done: youtube.com/watch?v=7DHE8RnsCQ8
$endgroup$
– stuart stevenson
Dec 29 '18 at 12:48
$begingroup$
When actually building a mobile you start with the bottom rods. Hang two weights at the ends of one, find the point on the rod where they balance (the center of gravity/mass). Then treat that rod+weights as if it werre just a weight. You can turn this into a set of linear equations.
$endgroup$
– Ethan Bolker
Dec 29 '18 at 13:08
add a comment |
$begingroup$
I've been interested for some time in hanging mobiles. (See Alexander Calder, for example)
To simplify, one can start by ignoring the mass of the rods and string, and only consider:
arm lengths
leaf masses
Doing this, we can define a mathematical 'mobile' M as an L-system:
$M -> M_l A_l A_r M_r$
$M -> L$
$A -> (arm) length$
$L -> (leaf) mass$
$l = left$
$r = right$
Where $M_l * A_l = A_r * M_r$ at every level.
This can lead to an interesting system of equations, since a mobile must always balance at every level within each subtree.
E.g.:
$M_{ll} * A_{ll} = A_{lr} * M_{lr}$,
where $ll$ means the left branch of the left branch, starting from the top, and $lr$ means the right branch of the left branch, etc.
But also there are inter-level dependencies of mass, since:
$M_l = M_{ll} + M_{lr}$
I am interested to understand how to model this system of equations and solve it. I see that it can be done with linear algebra, but don't quite know the next step, since the mass dependencies make things more complicated.
Also just ways to think about this system would be helpful.
This is for my own interest, fun, and learning, glad to learn with others.
Thanks for any help!
linear-algebra
asked Dec 29 '18 at 12:44
user7400474user7400474
11
$endgroup$
I've been interested for some time in hanging mobiles. (See Alexander Calder, for example)
To simplify, one can start by ignoring the mass of the rods and string, and only consider:
arm lengths
leaf masses
Doing this, we can define a mathematical 'mobile' M as an L-system:
$M -> M_l A_l A_r M_r$
$M -> L$
$A -> (arm) length$
$L -> (leaf) mass$
$l = left$
$r = right$
Where $M_l * A_l = A_r * M_r$ at every level.
This can lead to an interesting system of equations, since a mobile must always balance at every level within each subtree.
E.g.:
$M_{ll} * A_{ll} = A_{lr} * M_{lr}$,
where $ll$ means the left branch of the left branch, starting from the top, and $lr$ means the right branch of the left branch, etc.
But also there are inter-level dependencies of mass, since:
$M_l = M_{ll} + M_{lr}$
I am interested to understand how to model this system of equations and solve it. I see that it can be done with linear algebra, but don't quite know the next step, since the mass dependencies make things more complicated.
Also just ways to think about this system would be helpful.
This is for my own interest, fun, and learning, glad to learn with others.
Thanks for any help!
linear-algebra
linear-algebra
asked Dec 29 '18 at 12:44
user7400474user7400474
11
asked Dec 29 '18 at 12:44
user7400474user7400474
11
edited Dec 29 '18 at 13:16
user7400474
asked Dec 29 '18 at 12:44
user7400474user7400474
11
asked Dec 29 '18 at 12:44
user7400474user7400474
11
asked Dec 29 '18 at 12:44
user7400474user7400474
11
11
$begingroup$
Possibly related, A numberphile video in the context of balancing centrifuges and when this can be done: youtube.com/watch?v=7DHE8RnsCQ8
$endgroup$
– stuart stevenson
Dec 29 '18 at 12:48
$begingroup$
When actually building a mobile you start with the bottom rods. Hang two weights at the ends of one, find the point on the rod where they balance (the center of gravity/mass). Then treat that rod+weights as if it werre just a weight. You can turn this into a set of linear equations.
$endgroup$
– Ethan Bolker
Dec 29 '18 at 13:08
add a comment |
$begingroup$
Possibly related, A numberphile video in the context of balancing centrifuges and when this can be done: youtube.com/watch?v=7DHE8RnsCQ8
$endgroup$
– stuart stevenson
Dec 29 '18 at 12:48
$begingroup$
When actually building a mobile you start with the bottom rods. Hang two weights at the ends of one, find the point on the rod where they balance (the center of gravity/mass). Then treat that rod+weights as if it werre just a weight. You can turn this into a set of linear equations.
$endgroup$
– Ethan Bolker
Dec 29 '18 at 13:08
$begingroup$
Possibly related, A numberphile video in the context of balancing centrifuges and when this can be done: youtube.com/watch?v=7DHE8RnsCQ8
$endgroup$
– stuart stevenson
Dec 29 '18 at 12:48
$begingroup$
Possibly related, A numberphile video in the context of balancing centrifuges and when this can be done: youtube.com/watch?v=7DHE8RnsCQ8
$endgroup$
– stuart stevenson
Dec 29 '18 at 12:48
$begingroup$
When actually building a mobile you start with the bottom rods. Hang two weights at the ends of one, find the point on the rod where they balance (the center of gravity/mass). Then treat that rod+weights as if it werre just a weight. You can turn this into a set of linear equations.
$endgroup$
– Ethan Bolker
Dec 29 '18 at 13:08
$begingroup$
When actually building a mobile you start with the bottom rods. Hang two weights at the ends of one, find the point on the rod where they balance (the center of gravity/mass). Then treat that rod+weights as if it werre just a weight. You can turn this into a set of linear equations.
$endgroup$
– Ethan Bolker
Dec 29 '18 at 13:08
add a comment |
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$begingroup$
Possibly related, A numberphile video in the context of balancing centrifuges and when this can be done: youtube.com/watch?v=7DHE8RnsCQ8
$endgroup$
– stuart stevenson
Dec 29 '18 at 12:48
$begingroup$
When actually building a mobile you start with the bottom rods. Hang two weights at the ends of one, find the point on the rod where they balance (the center of gravity/mass). Then treat that rod+weights as if it werre just a weight. You can turn this into a set of linear equations.
$endgroup$
– Ethan Bolker
Dec 29 '18 at 13:08