Answer
$M=464N$
Work Step by Step
1) Find the angular acceleration of the dart
We know the initial tangential speed of the dart $v_0=0$, the final speed $v_T=5m/s$ and $t=0.1s$. The tangential acceleration of the dart is $$a_T=\frac{v_T-v_0}{t}=50m/s^2$$
The radius of rotation is $r=0.28m$. We can find the angular acceleration using $$\alpha=\frac{a_T}{r}=178.57rad/s^2$$
2) The torque required to help the dart to have $\alpha=178.57rad/s^2$ is $$\tau=I\alpha=(0.065kg.m^2)(178.57rad/s^2)=11.61N.m$$
This torque is produced by $M$ and has a lever arm of $0.025m$. Therefore, $$M=\frac{\tau}{0.025m}=464N$$
