1) For a rotating object, we can write the analogous of Newton's second law as
[tex]\tau = I \alpha[/tex]
where [tex]\tau[/tex] is the torque applied to the object, I is the moment of inertia and [tex]\alpha [/tex] is the angular acceleration.
2) We can write the ratio between the starting torque and the stopping torque as:
[tex] \frac{\tau _{start}}{\tau _{stop}} = \frac{I \alpha _{start}}{I \alpha _{stop}} = \frac{ \alpha _{start}}{\alpha _{stop}} [/tex]
3) Then, we can rewrite the two angular accelerations. The general formula is
[tex]\alpha = \frac{\omega _f - \omega _i}{t} [/tex]
where [tex]\omega _f [/tex] and [tex]\omega _i[/tex] are the final and initial angular velocities, while t is the time.
4) Let's start by calculating the starting acceleration. In this case, the initial velocity is zero, while the final velocity is the full rotational speed (let's call it [tex]\omega _{max}[/tex], and the time is t=5.0 s:
[tex]\alpha _{start} = \frac{\omega_{max}-0}{5 s}= \frac{\omega_{max}}{5 s}=[/tex]
When the carousel stops, instead, the initial velocity is [tex]\omega _{max}[/tex] while the final velocity is 0, and the time is t=11.0 s:
[tex]\alpha _{stop} = \frac{0-\omega _{max}}{11 s}= \frac{-\omega _{max}}{11 s} [/tex]
Therefore, if we replace the two angular accelerations that we found into the formula of the torque ration written at step 2), we find:
[tex] \frac{\tau _{start}}{\tau _{stop}} = - \frac{11}{5} [/tex]
where the negative sign simply means that the two torques have opposite direction. Therefore, the ratio between starting and stopping torque is 11:5.
