Pulling a wheel with a string?

A bicycle wheel is mounted on a fixed, frictionless axle, with a light string wound around its rim. The wheel has moment of inertia I = k m r^2, where m is its mass, r is its radius, and k is a dimensionless constant between zero and one. The wheel is rotating counterclockwise with angular velocity omega_0, when at time t=0 someone starts pulling the string with a force of magnitude F. Assume that the string does not slip on the wheel.

ASuppose that after a certain time t, the string has been pulled through a distance L. What is the final rotational speed omega_final of the wheel? Express answer in terms of L, F, I, and omega_0. I found that the final kinetic energy of the wheel is K_f = .5I omega_0^2 + FL, but I can't find an equation that relates this to final rotational speed.
BWhat is the instantaneous power P delivered to the wheel via the force F at time t = 0?

a)♣ the string being pulled,
the angular speed is: w1=w0 +w’t, hence t=(w1-w0)/w’;
the angular path is: b=w0t+0.5w’t^2, where angular acceleration
w’=T/J, torq T=Fr, and br=L;
♦ thus b=w0(w1-w0)/w’ +0.5w’((w1-w0)/w’)^2 =
= (w1-w0)(w0 +0.5w1 -0.5w0)/w’ =0.5(w1^2 –w0^2)/w’;
♠ and b=L/r =0.5(w1^2 –w0^2)/(Fr/J);
2(Fr/J)L/r =w1^2 –w0^2, hence
w1^2=2FL/J +w0^2;
b)♣ the power is P=F(w0r);

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