Equations of Motion for a mass on a circular track with a linear spring

[tlonster]

Homework Statement

A collar of mass m slides without friction along a circular track of radius R as
shown in THE ATTACHMENT. Attached to the collar is a linear spring with spring constant K and
unstretched length zero. The spring is attached at the fixed point A located a distance
2R from the center of the circle. Assume gravity acts down and determine (a) the differential equation of motion for the collar in terms the angle θ and (b) the reaction force exerted by the track on the collaras a function of the angle θ.

Homework Equations

The transport theorem

F = ma

The Attempt at a Solution

I have worked through this problem and am not confident in my answer. A second look would be much appreciated. Here's my answer:

I created reference frame E: {er, eθ, ez} at mass m such that it's position is

r = Rer

By using the transport theorem for velocity and acceleration as seen by the intertial frame I came up with:

v(m/I) = 0 + θ(dot)R[eθ]
a(m/I) = θ(double dot)R[eθ] - (θ(dot))^2R[er]

Next the forces acting on my mass are gravity, normal, and linear spring

∴Ʃ F = mg[JI] + N[er] + K(3R-0)[-er + JI]

where JI = sinθ[er] + cosθ[eθ]

when I substitute JI into my ƩF equation, set equal to ma(m/I), and drop the unit vectors, I get these two equations:

1. mgsinθ + N -3KR + 3KRsinθ = -m(θ(dot))^2R
2. mgcosθ + 3KRcosθ = mθ(double dot)R

For my equation of motion in terms of θ, I only need to use the second equation, correct? Where θ(double dot) = (mgcosθ + 3KRcosθ)/(mR)

and for the normal force (N) I just use the first equation and move everything to one side for N = -mgsinθ +3KR - 3KRsinθ -m(θ(dot))^2R

Can someone please take a look at this? I feel as though my spring force
Fs = K(3R-0)[-er + JI] is not correct, but I don't know what else it should be. Any help would be appreciated.

Thank you.

 

[KataKoniK]

Your solution looks correct to me. The only thing I would suggest is to double check your units and make sure they are consistent throughout the equations. Other than that, your approach and equations seem to be correct. Keep up the good work!