Equation of Motion for a Pendulum

[gumball]

Homework Statement

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.

Homework Equations

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

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

 

[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.

 

[gumball]

makes sense now, thanks for the help :D