Question

A spool with a thread wound on it is placed on an inclined smooth plane set at an angle $\alpha ={30}^{\circ }$ to the horizontal. The free end of the thread is attached to the wall as shown in figure above. The mass of the spool is $m=200g$, its moment of inertia relative to its own axis $I=0.45g{m}^2$, the radius of the wound thread layer $r=3.0cm$. If the acceleration of the spool axis is $x$, find the value of $10x$.

Answer 1

Let us depict the forces and their points of application for the spool. Choosing the positive direction for $x$ and $\phi$ as shown in the figure below, we apply $F_x=m{w}_{cx}$ and $N_{cz}=I_c{\beta }_z$ and get
$mg\sin \alpha -T=mw$; $Tr=I\beta$
"Notice that if a point of a solid in plane motion is connected with a thread, the projection of velocity vector of the solid's point of contact along the length of the thread equals the velocity of the other end of the thread (if it is not slacked)"
Thus in our problem, $v_p=v_0$ but $v_0=0$, hence point $P$ is the instantaneous centre of rotation of zero velocity for the spool. Therefore $v_c=\omega r$ and subsequently $w_c=\beta r$.
Solving the equations simultaneously, we get
$w=\frac{g\sin \alpha }{1+\frac{I}{m{r}^2}}=1.6m/s^2$