# Equation of Motion for a Pendulum

1. Aug 12, 2010

### gumball

1. The problem statement, all variables and given/known data

The question involves a simple pendulum, I am given three equations (1), (2) and (3) of motion for the bob at latitude (fi) for the x, y and z components.

the question then tells me to show that for small displacements meaning |theta|<< 1 (the angle between the string and the z direction in the centre of the x and y planes is very small) the three equations of motion will reduce to equations (4) (5) and (6).

my main problem is understanding how the equation in the z component reduces, especially regarding why the Tension remains as well as the force of gravity.

I also don't know the reason why we have to divide everything by mass.

2. Relevant equations

[PLAIN]http://img802.imageshack.us/img802/4642/equationsofmotion.jpg [Broken]

3. The attempt at a solution

I understand when (theta) is very small, the Tension in the string is almost equal to (mg)
T = mg

this explains that in the x and y components the Tension becomes mg. but in the z component I don't understand why Tension stays as T, while still keeping mg in the equation.

as for mass, Im guessing mass is negligible because the tension is mg, thus mass cancels out, but still doesn't explain why the z component stays.

Last edited by a moderator: May 4, 2017
2. Aug 12, 2010

### 6Stang7

Look up Small Angle Approximation and see if that helps.

The reason for dividing by mass is to get it into a standard differential equation forum.

3. Aug 13, 2010

### gumball

makes sense now, thanks for the help :D