Determine the acceleration and angular acceleration of a disc The question is: 

A 90kg disc is floating in a frictionless vacuum. A 150N force is
  applied to the outer rim of the disc. The disc has a radius of 0.25m
  and a radius of gyration of 0.16m. What is the acceleration and
  angular acceleration of this disc?

To solve it I set up these equations:
\begin{equation}
\mathbf{F}_{a} = \mathbf{m}\times\mathbf{a}\tag{1}
\end{equation}
\begin{equation}
\mathbf{F}_{\alpha} = \frac{\mathbf{I}\times\mathbf{\alpha}}{0.25}\tag{2}
\end{equation}
\begin{equation}
\mathbf{F}_{a}+\mathbf{F}_{\alpha} = 150 N\tag{3}
\end{equation}
You can find I and plug it in along with m, but that still leaves 4 unknowns and only 3 equations. I need a 4th equation but I'm not sure what else is known about the problem. 
 A: First, it's important to properly understand the equations of rotational motion. Rather than $F = ma,$ the operative equation of motion is $\tau = I \alpha$, where $\tau$ is the torque, $I$ is the moment of inertia and $\alpha$ is the angular acceleration. This problem also requires you to know the definition of the radius of gyration in the form of $r_g \equiv \sqrt\frac{I}{m}$. With a proper understanding of the definitions of $\tau$ and $\alpha$, you then have all the information that you need to solve the problem.
EDIT: The above answer referred solely to the rotational acceleration of the disk. The translational acceleration of the center of mass of the disk must be worked out separately using $F = ma$.
A: There is only one force applied to the system $F = 150\,{\rm N}$ not two, $F_a$ and $F_\alpha$.
Then you have 


*

*The total force applied equals the mass times the acceleration of the center of gravity.

*The total torque applied equals the mass moment of inertia (at the center of gravity) times the angular acceleration plus the gyroscopic forces (which are zero in your case).


So what is the torque applied when $F$ is located at the edge of the disk?
What is the mass moment of inertia of disk of mass $m$ with radius of gyration $\rho$?
Once you answer the above questions you can proceed to solve your problem.
