Particle moving on a conical surface

1. Dec 3, 2012

Drajcoshi

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

A particle moves under the action of gravity on a conical surface z^2 = 4(x^2+ y^2),
z ≥ 0, where z is the vertical axis. For initial position r = (1, 0, 2) and initial velocity
ṙ = (0, 2, 0) find the extremal values of z along the trajectory. Take g = 10.

2. Relevant equations

I really have not a clue how to type the equation on this site but have uploaded the work out on pdf. will appreciate if anyone can shed some light on this. thanks

Attached Files:

• particle on a conical surface.pdf
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Last edited by a moderator: Dec 4, 2012
2. Dec 3, 2012

TSny

Hello, Drajcoshi. Welcome to Physics Forums!

I'm not following your set up of polar coordinates. Did you really want to set $z = \rho$?

I think you can solve this problem with just application of conservation laws. Besides energy, can you think of anything else that's conserved?

3. Dec 3, 2012

Drajcoshi

At this point I am we'll confused, how would you do it? Really appreciate your help. I think I completely messed up the calculation.

4. Dec 3, 2012

TSny

There's another quantity that's conserved (hint: it's the z-component of some vector quantity).

With this quantity and energy you will be able to set up equations to determine max or min of z.

5. Dec 4, 2012

Drajcoshi

alright, did more calcultion and stuck dont know how to find the max and min of z. I have included the calculation, could you check it for me? thanks

https://www.dropbox.com/s/0qq6cyopw4d0bvj/photo.JPG

6. Dec 4, 2012

TSny

OK, so you have that the z-component of angular momentum as well as the total energy is conserved.

Try writing expressions for $L_z$ and $E$. You are only concerned with points of max or min $z$ and the expression for $L_z$ will simplify at those points.

7. Dec 4, 2012

Drajcoshi

this this correct? what happens next? sorry all this maths notation is so confusing since i am not a maths student. also dont know how to write using the equation editor.

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8. Dec 4, 2012

TSny

I'm not really following your notation. If you let $v$ represent the speed of the particle, how would you express $E$ in terms of $v$ and $z$?

Can you also express $L_z$ in terms of $v$ and $z$ at a point of maximum or minimum $z$? (First express it in terms of $v$ and $\rho$ and then express $\rho$ in terms of $z$.)