Answer
The mass of the wheel is 46.5 kg.
Work Step by Step
We can find the speed $v$ of the object after it falls 3.00 m.
$\frac{v+v_0}{2} = \frac{y}{t}$
$v = \frac{2y}{t} = \frac{(2)(3.00~m)}{2.00~s}$
$v = 3.00~m/s$
We can find the kinetic energy of the object.
$KE = \frac{1}{2}mv^2$
$KE = \frac{1}{2}(4.20~kg)(3.00~m/s)^2$
$KE_{object} = 18.9~J$
We can find the kinetic energy of the wheel.
$KE_{wheel} = mgh - KE_{object}$
$KE_{wheel} = (4.20~kg)(9.80~m/s^2)(3.00~m) - 18.9~J$
$KE_{wheel} = 104.58~J$
We can find the mass $M$ of the wheel.
$KE_{wheel} = \frac{1}{2}I\omega^2 = 104.58~J$
$\frac{1}{2}(\frac{1}{2}MR^2)(\frac{v}{R})^2 = 104.58~J$
$M = \frac{(4)(104.58~J)}{v^2}$
$M = \frac{(4)(104.58~J)}{(3.00~m/s)^2}$
$M = 46.5~kg$
The mass of the wheel is 46.5 kg.
