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Equations of motion

Thread starter gtk
Start date Nov 13, 2016
Tags

Equations of motion Free body diagram Hard Motion Polar coordinates Time Writing

1 reply · 2K views · last reply Nov 13, 2016

gtk

Thread starter
Nov 13, 2016

#1

Homework Statement

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3D sketch of a mass m resting on a saddle-shaped curve in the xy-plane with z vertical, the curve rising on both sides of the origin; angle theta is marked from the x-axis to the mass's projection on the plane.

Typed text: Particle of mass m moving on frictionless track given by z = αr² − βr⁴, where r² = x² + y², track lies in a vertical plane driven about the z axis with constant angular velocity θ̇ = Ω, trying to write equation of motion in polar cylindrical coordinates from F = ma.

Homework Equations

The Attempt at a Solution

Handwritten derivation in cylindrical polar basis showing position r = r e_r + (αr² − βr⁴) e_z, then successive time derivatives of velocity and acceleration with terms in Ω, α and β, ending with r̈ = (r̈ − rΩ²) e_r + 2ṙΩ e_θ + (2α − 12βr²) r̈ e_z.

Free body diagram of a bead on a straight rod inclined in the plane: red arrows show a force -mg (dz/dr)(dr/dt) e_r along the rod's radial direction, F_theta e_theta pointing right, and -mg e_z pointing downward.

Handwritten derivation in cylindrical coordinates: F = m r'' expanded into force components, with e_r, e_theta and e_z balance equations labelled Equation of Motion and Reaction Force.

Typed text reading: Is this even close to being correct? I think the e_r EOM LHS does not even have the right units: e_r : -mg(2αr - 4βr^3)ṙ = That's not a force.... Right?.... man, I'm terrible at physics.

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Discussion

gtk

Nov 13, 2016

#2

I changed my FBD, to make the units correct. But I still don't know if this is correct.

Free body diagram of a bead on a rotating straight wire, with a gray circle on a black line. Labeled forces: -mg (dz/dr) e_r pointing left along the wire, F_theta e_theta out of the page, and -mg e_z pointing down.

My new EOM and reaction forces come out to:

Three equations in cylindrical components: e_r: -mg(2αr - 4βr^3) = m(r̈ - rΩ^2) labeled Equation of Motion; e_θ: F_θ = 2mṙΩ labeled Reaction Force; e_z: -mg = m(2α - 12βr^2)r̈ labeled Reaction Force.

Is this correct? Or did I mess something up?

Thanks!

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