So, say you have a free to rotate disc, assuming no external torques, and you have a spool, radius 7.93 mm, attached to its centre.
Say the spool has a string attached to a point on its edge and the string supports a mass, 14.7 g.
Now you wind up the string around the spool and allow the mass to fall, accelerating at linear acceleration $a_l$ and causing the disc to rotate at angular acceleration 0.66 rad/s.
How is the angular acceleration of the disc related to linear acceleration of the mass? And how would I use the angular acceleration to find the $a_l$? In other words by what equation would I be able to find the $a_l$ from the angular acceleration of the disc?
Does it have something to do with the fact that each time the spool rotates one revolution, its position has changed by $2\pi r$, so therefore, the amount of string fed out would equal $2\pi r$?
If so, how would I convert that information into an equation relating the two accelerations?
Best Answer
You know the string length is constant so a small change in angle $\Delta\theta$ on the pulley yields what change in height $y$ of the mass?
once you have $\Delta y = K \Delta \theta$ you can differential twice to get
$$ \frac{{\rm d} y}{{\rm d}t} = K \frac{{\rm d} \theta}{{\rm d}t} \\ \frac{{\rm d}^2 y}{{\rm d}t^2} = K \frac{{\rm d}^2 \theta}{{\rm d}t^2} $$
since coefficient (gearing) $K$ you come up with is constant with time. The above is simply $a = K \alpha$ relating the linear to the angular acceleration.