How Do You Solve Particle Motion on a Horizontal Table with a Descending Cord?

[RedRollins]

Homework Statement

A particle P of mass m moves on a smooth, horizontal table, and is attached to a light, inextensible cord that is being pulled downward by a force F(t) as shown. The differential equations of motion are
-T = m(r(double dot) - r*(theta(dot))^2)
0 = r*(theta(doubledot)) + 2*r(d0t)*theta(dot)

Let the particle be at r = r(knot) at t = 0, and let the part of the cord beneath the table be descending at constant speed vc. If the transverse component of velocity of P is r*theta(dot) = r(knot)*theta(knot)

The Attempt at a Solution

I realize that the problem implies that r^2*theta(dot) = a constant and since its traveling at a constant speed r(dot) = vc but I'm just confused as to what to do
If someone can just give me an idea of where to go. A lot less concerned with getting the right answer then how.
(just 78 not 77 in the pictures)

 

[nilesh_pat]

So for the differential equations, you need to solve them using the given initial conditions. This means that you will need to use the initial conditions to find the constants of integration, and then solve the differential equations for r and θ.For the first equation, you can use the initial condition to find the constant of integration, which is m*(r_knot-vc)*theta_knot^2. Then you can solve the equation for r, and obtain the expression r(t) = vc*t + r_knot - m*theta_knot^2/vc.For the second equation, you can use the initial condition to find the constant of integration, which is -2*m*r_knot*theta_knot. Then you can solve the equation for θ, and obtain the expression θ(t) = (1/2)*vc*t^2 + r_knot*theta_knot/vc - m/(vc^2)*ln(1+vc*t/r_knot).Hope this helps!