The mass m is suspended by a string from a disk circular homogeneous of mass mc and radius R, as illustrated in figure 24. The disk is restricted from rotating by a spring attached to it at a distance cr of its center of rotation. If mass m undergoes a downward displacement x, determine the frequency of oscillation of the disk.
Can someone explain to me why mx and Jteta are in that direction (have positive sinal) ?
If we set x down as positive, $j \ddot\theta$ and $kyR_c$ should have negative signs.
Because when the mass m accelerates down, those two resist its acceleration.
The choice of:
The most common problem in dynamics problems is what happens with derived quantities e.g. moment of an unknown force. More specifically, if it required to determine the moment of a $m\ddot{x}$ term (which describes an "active" force), then depending on the position of the $m\ddot{x}$ relative to the point that the moments are calculated the resulting moment can have a positive or negative sign.
An additional example --which commonly produces confusion -- is the term $J \cdot \ddot{\theta}$. Although the positive direction for $\theta$ is defined as ClockWise(CW), the moment is positive for CounterClockWise (CCW). So in this case, I suspect what is happening is that whoever drew the diagram drew the term $J \cdot \ddot{\theta}$ as positive with respect to the moment positive.
Apart from that I am not certain that in this problem you can ignore gravity. Which means that the force on the rope will not be equal to $m\cdot \ddot{x}$ (it can actually be derived by $m\cdot g -T = m\cdot \ddot{x}$).
Also, I agree with you that $\ddot{x}$ should be pointing downwards, if $\theta $is rotating counterclockwise (unless $\dot{\theta}$ is too large).
However, in general it will be very difficult to decypher what was the exact methodology that the person used to derive the equation (or his/her intent),without a more detailed explanation of the solution. .
