Does a cylinder/cone being spun around on an arm have angular momentum? This question comes after seeing a video about the SpinLaunch company and their "centrifuge" launch system.  As shown in the basic illustration below, a "rocket" shaped projectile will be spun around at very high speeds and then released.
Since the projectile is being held so that it's always facing tangent to the circle, is a torque being applied to the projectile?  If a torque is being applied, does that mean that when it's released, it will want to keep spinning?  I feel like this might depend on the arm being attached to the exact center of gravity of the projectile, but I can't put all the pieces together in my mind.
This is similar to a hammer throw, except the hammer is thrown with the string/chain attached which stops the ball from spinning (if it was spinning).  So I guess this can be rephrased as, if the ball was released from the chain (rocket detaching from the arm), would the ball be spinning around its own axis as it flew off tangent to its rotation around the athlete?

 A: Yes if the projectile were accelerated in the way illustrated, it is going to be rotating around its own center of mass at the same rate that the rotating arm is spinning.
When released, it will want to maintain that rotation unless a torque is applied to stop it.
An answer over on Space SE suggests that (in the case of SpinLaunch) the release mechanism will be used to apply such torque.
A: The rocket will be spinning after launch, if the contact arm is suddenly detached from the COM when it is in a vertical position.
We at first think that the rocket at that instant has a defined upwards velocity and there is no reason for the spin. But the angular velocity is the same for all rocket points. For the COM, the upwars velocity is $\mathbf v_c = \boldsymbol \omega \times \mathbf r_c$. For the tail, $\mathbf v_t = \boldsymbol \omega \times \mathbf r_t$.
$\mathbf r_c$ and $\mathbf r_t$ are different vectors with different directions. The same then for $v_c$ and $v_t$. The conclusion is that the rocket is spinning around its COM at the launch instant, and will keep spinning if some countermeasure is not used.
