Final answer:
A member fixed to a collar on a smooth rod would exhibit translational motion as it can move along the rod without rotation. There is no indication of a restoring force that might cause oscillatory motion, nor is there a central pivot point for rotational motion implied by the question.
Step-by-step explanation:
A member fixed to a collar on a smooth rod would exhibit translational motion. This is because the nature of the connection implies that the member can move along the rod, but without rotating around its own axis or the rod. To analyze the motion of the member, one must consider that in a pure translational motion scenario, all points on the object move in parallel paths, implying that there is no rotation about any point within the object, or about an external pivot point such as a rod.
However, if we consider the possibility of oscillatory motion, we can recall that such motion would be apt for a system described by Hooke's law, typically referred to as a simple harmonic oscillator. Since the question mentions the rod is smooth and does not mention a restoring force that would cause oscillatory motion, it is less likely the member would exhibit this type of motion.
If the collar and rod system was part of a rotary device and constrained to move in uniform circular motion, then the member would rotate around a central point. But given the simplicity of the setup as described, where the collar presumably allows the member to slide without friction, the most straightforward conclusion to expect is translational motion.