What Is the Correct Equation of Motion for a Pendulum at Any Amplitude?

[olgerm]

Homework Statement

what is equation of motion for pendulum?pendulum is made of pointmass, which mass is m, is fixed to thread ,which length is l? Oscillation aplitude is θ.Other side of thread is fixed in (0;0;0)point. at time t=0
$t=0;y=0<br /> ;z=0<br /> ;φ=θ$

Homework Equations

$\frac{dφ^2}{dt^2}=-\frac{g}{l}*Sin(φ)$

The Attempt at a Solution

x(t)=?
y(t)=0
z(t)=0
t(t)=t

 

[Dr. Courtney]

Small angle approximation!

 

[olgerm]

Homework Statement

what is equation of motion for pendulum?pendulum is made of pointmass, which mass is m, is fixed to thread ,which length is l? Oscillation aplitude is θ.Other side of thread is fixed in (0;0;0)point. at time t=0
$t=0;<br /> ;z=0<br /> ;φ=θ$

Homework Equations

$\frac{dφ^2}{dt^2}=-\frac{g}{l}*Sin(φ)$

The Attempt at a Solution

x(t)=?
y(t)=?
z(t)=0
t(t)=t

 

[olgerm]

Small angle approximation!

What is small angel approximation ? The relevant equation I wrote?

 

[Dr. Courtney]

What is small angel approximation ? The relevant equation I wrote?

Google it.

 

[HallsofIvy]

Draw a picture! If the line of the pendulum is at angle $\theta$ to the vertical the force acting on the pendulum bob is straight down but the pendulum string prevents the bob from moving straight down. Divide the force into components perpendicular to and parallel to the circular arc the pendulum bob makes. The use "force= mass times acceleration".

 

[olgerm]

$φ=θ*sin(\sqrt{\frac{g}{l}*t})$

$\begin{cases}<br /> x=sin(φ)*l\\<br /> y=l*(cos(φ)*-1)\\<br /> \end{cases}$

So correct equation of motion is
$\begin{cases}<br /> x=sin(θ*sin(\sqrt{\frac{g}{l}*t}))*l\\<br /> y=cos(θ*sin(\sqrt{\frac{g}{l}*t}))*l-l\\<br /> \end{cases}$
?

 

[haruspex]

$φ=θ*sin(\sqrt{\frac{g}{l}*t})$

$\begin{cases}<br /> x=sin(φ)*l\\<br /> y=l*(cos(φ)*-1)\\<br /> \end{cases}$

So correct equation of motion is
$\begin{cases}<br /> x=sin(θ*sin(\sqrt{\frac{g}{l}*t}))*l\\<br /> y=cos(θ*sin(\sqrt{\frac{g}{l}*t}))*l-l\\<br /> \end{cases}$
?

For small amplitude, that's roughly right, but doesn't satisfy the given initial conditions.
However, the OP does not specify small angles, so it's not clear whether this is what is wanted. Maybe they just want the differential equation, but using x and y instead of $\phi$.

 

[olgerm]

For small amplitude, that's roughly right, but doesn't satisfy the given initial conditions.
However, the OP does not specify small angles, so it's not clear whether this is what is wanted. Maybe they just want the differential equation, but using x and y instead of $\phi$.

What is the correct equation for any amplitude?
I mean motion of equation of pendulum "head".
x(t)=??
y(t)=??

 

[haruspex]

What is the correct equation for any amplitude?
I mean motion of equation of pendulum "head".
x(t)=??
y(t)=??

There is no analytic solution.