Answer
The angular speed is 551 rpm.
Work Step by Step
We can find the speed of the string after 1.0 meter of string has been unwound:
$v^2 = v_0^2+2ax = 0 +2ax$
$v =\sqrt{2ax} = \sqrt{(2)(1.5~m/s^2)(1.0~m)}$
$v = 1.73~m/s$
We can find the angular speed;
$\omega = \frac{v}{r} = \frac{1.73~m/s}{0.030~m}$
$\omega = 57.7~rad/s$
We can express the angular speed in units of rpm:
$\omega = (57.7~rad/s)(\frac{60~s}{1~min})(\frac{1~rev}{2\pi~rad})$
$\omega = 551~rpm$
The angular speed is 551 rpm.
