Circular motion velocity components problem

AI Thread Summary
The discussion revolves around a circular hoop with a radius R and mass M, which is initially spinning without linear motion. When an object of mass m is fired from a spring-loaded gun at an angle theta, the focus shifts to calculating the x and y components of the center of mass velocity post-launch. Participants highlight the conservation of momentum as a key principle, noting that while the hoop has angular momentum, it lacks linear momentum. Clarifications are sought regarding the terminology used, particularly the description of the center of mass and its relationship to the hoop and the object. The conversation emphasizes the need for precise definitions to solve the problem effectively.
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Consider a circular hoop with a radius R and a mass M. Initially the centre of mass of the hoop is spinning, but not moving. Then a object with mass m is thrown at speed v by a spring-loaded gun with an angle of theta anticlockwise from the posive horizontal. Calculate the x and y components of the centre of mass velocity after the object is thrown.

So this looks like a conservation of momentum problem. Since the CM is spinning but not moving, it should have angular momentum but not linear momentum. But how do I calculate it with just R and M? The components of the object's velocity are easy to calculate using theta. But where to from there?
 
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I do not understand the description. First of all, the CM cannot be spinning. It is a point. Did you mean to say the hoop is spinning?

Secondly, what is the connection between the object, the gun and the hoop? What CM is being used in the final part of the description?
 

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