A particle of mass m slides (both sideways and radially) on a smooth frictionless horizontal table. It is attached to a cord that is being pulled downwards at a prescribed constant speed v by a force T (T may be varying)
Use F=ma in polar coordinates to derive an expression for the tension T (T will depend on r and θ and how they may be changing)
Show the particle's polar coordinates satisfy r2dθ/dt = constant
HINT: The only horizontal force on the particle is T and it acts purely in the inward radial direction. Also, dr/dt is known and it is equal to -v
a = (r''(t)w2)r_hat +(r(t)θ''(t) + 2r'(t)θ'(t))θ_hat
L(t) = L_o ????
The Attempt at a Solution
Not entirely sure how to get started. I've identified that a_r must be zero (since dr/dt is a constant, r''(t) must be zero making a_r zero). I've also set up the equation a_θ = r(t)θ''(t) - 2vw since r'(t) = v and θ'(t) = w. The issue I'm having is trying to identify another equation for the acceleration (or force) in the θ direction, and then from there rectifying that into T.