Lagrangian mechanics: Kinetic energy of a bead sliding along a bent wire

[wdednam]

Homework Statement

Determine the kinetic energy of a bead of mass m which slides along a frictionless wire bent in the shape of a parabola of equation y = x². The wire rotates at a constant angular velocity $$\omega$$ about the y-axis.

Homework Equations

T = $$\frac{1}{2}$$m($$\dot{x}^2$$ + $$\dot{y}^2$$ + $${x}^2$$$$\omega^2$$)

The Attempt at a Solution

The above equation represents my attempt to write down the kinetic energy of the system in an appropriate coordinate system. After this I eliminated $$\dot{y}$$ in favour of $$\dot{x}$$ using y = x² and got:

T = $$\frac{1}{2}$$m($$\dot{x}^2$$ + 4$${x}^2$$$$\dot{x}^2$$ + $${x}^2$$$$\omega^2$$)

Does this look right to anyone? The book (study guide) I'm using was unfortunately compiled by my University and no answers are supplied to end-of-chapter problems. This problem comes out of the first chapter of my study guide and all the problems there basically involves writing down a correct expression for the Lagrangian/Kinetic Energy.

Thanks in advance for any help.

 

[turin]

Determine the kinetic energy of a bead of mass m which slides along a frictionless wire bent in the shape of a parabola of equation y = x². The wire rotates at a constant angular velocity $$\omega$$ about the y-axis.

T = $$\frac{1}{2}$$m($$\dot{x}^2$$ + 4$${x}^2$$$$\dot{x}^2$$ + $${x}^2$$$$\omega^2$$)

Does this look right to anyone?

It looks right to me. I am, of course, assuming, as you have apparently done as well, that x is the distance from the y-axis, and not simply the Cartesian x-coordinate.

 

[wdednam]

It looks right to me. I am, of course, assuming, as you have apparently done as well, that x is the distance from the y-axis, and not simply the Cartesian x-coordinate.

Hi Turin,

Thanks a lot for the help!

Yes, x does represent the distance from the y-axis, but I'm wondering if it wouldn't have been better to use cylindrical coordinates, z and r for y and x respectively, instead?

 

[turin]

Thanks a lot for the help!

I didn't even do anything, but, you're welcome. :)

... I'm wondering if it wouldn't have been better to use cylindrical coordinates, z and r for y and x respectively, instead?

Of course, those are just letters, and what we have both implicitly assumed is that these ARE, in fact, cylindrical coordinates (in disguise), in the way that you have identified. I don't know why the author decided to use those letters as opposed to the standard \rho and z, but that was the author's decision, not ours.