Answer
(a) The diameter of the disk is 7.36 meters.
(b) $a_c = 327~m/s^2$
Work Step by Step
(a) $M = \rho \times volume$
$M = \rho \times area \times (0.100~m)$
$M = (\rho)(\pi~R^2)(0.100~m)$
We can find the angular speed of the disk.
$\omega = (90.0~rpm)(\frac{1~min}{60~s})(2\pi)$
$\omega = 3\pi~rad/s$
We can find the radius $R$ of the disk.
$KE = 10.0\times 10^6~J$
$\frac{1}{2}I\omega^2 = 10.0\times 10^6~J$
$\frac{1}{2}(\frac{1}{2}MR^2)\omega^2 = 10.0\times 10^6~J$
$\frac{1}{4}(\rho)(\pi~R^2)(0.100~m)R^2\omega^2 = 10.0\times 10^6~J$
$R^4 = \frac{40.0\times 10^6~J}{(\rho)(\pi)(0.100~m)\omega^2}$
$R = (\frac{40.0\times 10^6~J}{(7800~kg/m^3)(\pi)(0.100~m)(3\pi~rad/s)^2})^{1/4}$
$R = 3.68~m$
The diameter of the disk is twice the radius. Thus the diameter of the disk is 7.36 meters.
(b) $a_c = \omega^2~R$
$a_c = (3\pi~rad/s)^2(3.68~m)$
$a_c = 327~m/s^2$
