How to find the tension of a wire tied to a ring which is going down?

Jun 2017
399
6
Lima, Peru
The problem is as follows:

A ring which its mass is $2\,kg$ and radius $R$ has a thin wire of negligible mass rolled in. In one end the wire is tied to the ceiling of a room. The ring which is connected to the wire falls vertically as it is depicted with no slippage. Given these conditions find the acceleration in meters per second square of the center of mass and the tension of the wire. Assume $g=10\,frac{m}{s^2}$



$\begin{array}{ll}
1.&5\frac{m}{s^2}\,,10\,N\\
2.&10\frac{m}{s^2}\,,20\,N\\
3.&5\frac{m}{s^2}\,,5\,N\\
4.&4\frac{m}{s^2}\,,8\,N\\
5.&2\frac{m}{s^2}\,,6\,N\\
\end{array}$

I'm confused exactly what should be the mass of this object in the center?. I'm assuming that it might imply that in the center for this object is $\frac{mg}{2}$?. But I'm confused exactly if this is the appropiate way to solve this problem. Can someone help me here please?.
 

skeeter

Math Team
Jul 2011
3,356
1,848
Texas
sketch a FBD

in the vertical direction ...

$mg - T = ma$

net torque ...

$T \cdot r = I\alpha = mr^2 \cdot \dfrac{a}{r} \implies T = ma$


substitute $ma$ for $T$ in the first equation ...