Horizontal and vertical independance

[phlegmy]

hey guys,
i studied "applied maths" in "high school" [the irish version!]
we studied projectile motion and velocity vectors etc,
we worked on the premise that gravity affects only the vertical velocity of the projectile
so that:
for a projectile given an initial velcoity both horizontally and vertically (i and j)
the horizontal component would remain unchanged [obviously diregarding any frictional forces], while the vertical component is subject to change due to gravity.
fine.

now i turn my mind to a pendulum, free to swing both left and right, as well as front to back, and i figure: the left-right motion can be treated seperatly to the front-back motion, each discribing, in that direction,the motion it would otherwise assume were the other direction neglected. and then:

i think surely not!
> i imagine that, in the left-right motion the pendulum is swinging not as a pendulum but rather in full circuls, and very fast. i then reason (rightly or wrongly) that were i to nudge the pendulum in the forward-backward direction that it would not oscillate forwards and backwards but rather be restored to its inital position. similar to the way that a bullet which spins along the axis of its trajectory will not become unstable.

assuming that what i have said is correct [open to correction!] i am left with a conumdrum:
when is it acceptable to split an objects motion into x,y or x,y,z and why
is it something to do with rotation?
or am i missing something else more fundamental.

thanks for your thoughts!
james

 

[Mentz114]

Hi James,
The difference between your two examples, the projectile and the pendulum, is crucial. In the projectile case only changes in the vertical direction change the energy of the projectile. In the case of the pendulum, movement in either direction will change the energy, because it is constrained to move in a circle. So x and y and not independent in the second case, but are so in the first.

You can see this mathematically if you write the Lagrangians of the two systems.

 

[phlegmy]

thats mentz, although you replied in less than an hour and its taken me 10 days to read it!
i see your point.