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.
Best Answer
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 $F = ma$.