# Pendulum: equation of motion

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1. Jul 25, 2015

### olgerm

1. The problem statement, all variables and given/known data
what is equation of motion for pendulum?pendulum is made of pointmass, which mass is m, is fixed to thread ,which lenght is l? Oscillation aplitude is θ.Other side of thread is fixed in (0;0;0)point. at time t=0
$t=0;y=0 ;z=0 ;φ=θ$

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

3. The attempt at a solution
x(t)=???
y(t)=0
z(t)=0
t(t)=t

2. Jul 25, 2015

### Dr. Courtney

Small angle approximation!

3. Jul 26, 2015

### olgerm

1. The problem statement, all variables and given/known data
what is equation of motion for pendulum?pendulum is made of pointmass, which mass is m, is fixed to thread ,which lenght is l? Oscillation aplitude is θ.Other side of thread is fixed in (0;0;0)point. at time t=0
$t=0; ;z=0 ;φ=θ$

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

3. The attempt at a solution
x(t)=???
y(t)=???
z(t)=0
t(t)=t

4. Jul 26, 2015

### olgerm

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

5. Jul 26, 2015

### Dr. Courtney

6. Jul 26, 2015

### HallsofIvy

Staff Emeritus
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".

7. Jul 26, 2015

### olgerm

$φ=θ*sin(\sqrt{\frac{g}{l}*t})$

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

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

8. Jul 28, 2015

### haruspex

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 $\phi$.

9. Jul 28, 2015

### olgerm

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

Last edited: Jul 28, 2015
10. Jul 28, 2015

### haruspex

There is no analytic solution.