Tipping ladder equation
$begingroup$
I try to solve a variant of the falling ladder problem, this time without a wall and the bottom of the ladder does not slide. There is a mass $m$ and the angle of the ladder with the vertical is $phi$.
The rotating moment is caused by the gravitational force:
$$M_r = F_z l sin phi = m g l sin phi $$
Which equals the rotational inertial moment:
$$M_phi = ml^2 frac{d^2 phi}{dt^2}$$
This results in the following differential equation:
$$frac{d^2 phi}{dt^2} = frac{g}{l} sin phi$$
Any idea how to solve this?
ordinary-differential-equations boundary-value-problem
asked Jan 6 at 20:26
APIAPI
1063
$endgroup$
add a comment |
$begingroup$
I try to solve a variant of the falling ladder problem, this time without a wall and the bottom of the ladder does not slide. There is a mass $m$ and the angle of the ladder with the vertical is $phi$.
The rotating moment is caused by the gravitational force:
$$M_r = F_z l sin phi = m g l sin phi $$
Which equals the rotational inertial moment:
$$M_phi = ml^2 frac{d^2 phi}{dt^2}$$
This results in the following differential equation:
$$frac{d^2 phi}{dt^2} = frac{g}{l} sin phi$$
Any idea how to solve this?
ordinary-differential-equations boundary-value-problem
asked Jan 6 at 20:26
APIAPI
1063
$endgroup$
1
$begingroup$
Solving this differential equation as it's written requires some special functions. Typically you'd want to linearize it for small displacement by $sin phi sim phi$ and solve that.
$endgroup$
– AEngineer
Jan 6 at 21:01
add a comment |
$begingroup$
I try to solve a variant of the falling ladder problem, this time without a wall and the bottom of the ladder does not slide. There is a mass $m$ and the angle of the ladder with the vertical is $phi$.
The rotating moment is caused by the gravitational force:
$$M_r = F_z l sin phi = m g l sin phi $$
Which equals the rotational inertial moment:
$$M_phi = ml^2 frac{d^2 phi}{dt^2}$$
This results in the following differential equation:
$$frac{d^2 phi}{dt^2} = frac{g}{l} sin phi$$
Any idea how to solve this?
ordinary-differential-equations boundary-value-problem
asked Jan 6 at 20:26
APIAPI
1063
$endgroup$
I try to solve a variant of the falling ladder problem, this time without a wall and the bottom of the ladder does not slide. There is a mass $m$ and the angle of the ladder with the vertical is $phi$.
The rotating moment is caused by the gravitational force:
$$M_r = F_z l sin phi = m g l sin phi $$
Which equals the rotational inertial moment:
$$M_phi = ml^2 frac{d^2 phi}{dt^2}$$
This results in the following differential equation:
$$frac{d^2 phi}{dt^2} = frac{g}{l} sin phi$$
Any idea how to solve this?
ordinary-differential-equations boundary-value-problem
ordinary-differential-equations boundary-value-problem
asked Jan 6 at 20:26
APIAPI
1063
asked Jan 6 at 20:26
APIAPI
1063
asked Jan 6 at 20:26
APIAPI
1063
asked Jan 6 at 20:26
APIAPI
1063
asked Jan 6 at 20:26
APIAPI
1063
1063
1
$begingroup$
Solving this differential equation as it's written requires some special functions. Typically you'd want to linearize it for small displacement by $sin phi sim phi$ and solve that.
$endgroup$
– AEngineer
Jan 6 at 21:01
add a comment |
1
$begingroup$
Solving this differential equation as it's written requires some special functions. Typically you'd want to linearize it for small displacement by $sin phi sim phi$ and solve that.
$endgroup$
– AEngineer
Jan 6 at 21:01
1
1
$begingroup$
Solving this differential equation as it's written requires some special functions. Typically you'd want to linearize it for small displacement by $sin phi sim phi$ and solve that.
$endgroup$
– AEngineer
Jan 6 at 21:01
$begingroup$
Solving this differential equation as it's written requires some special functions. Typically you'd want to linearize it for small displacement by $sin phi sim phi$ and solve that.
$endgroup$
– AEngineer
Jan 6 at 21:01
add a comment |
1 Answer
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This is the simple pendulum equation. Unfortunately, there is no closed form solution (except for the trivial $varphi=0$ unstable ladder that never slides).
answered Jan 7 at 4:30
Stefan LafonStefan Lafon
3,015212
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add a comment |
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$begingroup$
This is the simple pendulum equation. Unfortunately, there is no closed form solution (except for the trivial $varphi=0$ unstable ladder that never slides).
answered Jan 7 at 4:30
Stefan LafonStefan Lafon
3,015212
$endgroup$
add a comment |
$begingroup$
This is the simple pendulum equation. Unfortunately, there is no closed form solution (except for the trivial $varphi=0$ unstable ladder that never slides).
answered Jan 7 at 4:30
Stefan LafonStefan Lafon
3,015212
$endgroup$
add a comment |
$begingroup$
This is the simple pendulum equation. Unfortunately, there is no closed form solution (except for the trivial $varphi=0$ unstable ladder that never slides).
answered Jan 7 at 4:30
Stefan LafonStefan Lafon
3,015212
$endgroup$
This is the simple pendulum equation. Unfortunately, there is no closed form solution (except for the trivial $varphi=0$ unstable ladder that never slides).
answered Jan 7 at 4:30
Stefan LafonStefan Lafon
3,015212
answered Jan 7 at 4:30
Stefan LafonStefan Lafon
3,015212
answered Jan 7 at 4:30
Stefan LafonStefan Lafon
3,015212
answered Jan 7 at 4:30
Stefan LafonStefan Lafon
3,015212
3,015212
add a comment |
add a comment |
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Solving this differential equation as it's written requires some special functions. Typically you'd want to linearize it for small displacement by $sin phi sim phi$ and solve that.
$endgroup$
– AEngineer
Jan 6 at 21:01