Can a Bead on a Parabolic Wire Exhibit Simple Harmonic Motion?

[Mattofix]

Homework Statement

i have a bead on a wire shaped by y=cx^2 where y is the height - use conservation of energy to show that the system unfergoes simple harmonic motion. - no friction!

Homework Equations

I think to prove its SHM i need to get an equation in the form acc= - w^2 x

The Attempt at a Solution

KE + PE = A constant
1/2mv^2 + mgh = A constant
1/2mv^2 + mgcx^2 = A constant

... is this right? i am pretty sure i need it in the above form them from that i know what w is and can work out the period of this moyion.

 

[Mattofix]

please help - with this i can't do the rest of the question :s

 

[P3X-018]

Well as you said above, you need to show that the system satisfy an equation of the form acc = - constant * position.
More precisly something saying Z'' = - constant * Z. It doesn't matter what this constant exactly is, besides being a constant.
The v in your equation from the the kinetic energy is the SPEED of the bead so it's not the derivative of your x. What is v then? What happens if you plug that expression for v into your equation:

1/2mv^2 + mgcx^2 = constant

then taking the time derivative, so get something with dobbel derivative which is what you are seeking.

 

[Mattofix]

what do i plug in for v? I am totally confused...

 

[P3X-018]

It's the length of the velocity vector, so it's
v^2 = x'^2 + y'^2
where x' and y' are the velocities in the x- and y- direction respectivly (ie time derivatives of x and y resp.). What is y' in this case?

 

[Mattofix]

y' = 2xcx' ?

 

[Mattofix]

ahhh -= please help somebaody - its in for tomrrow

 

[P3X-018]

y' = 2xcx' ?

This is correct.