Common classroom examples of uniform circular motion include, for obvious reasons, the simplest model: a ball tied to a string being swung in a horizontal circle. Since the net force is in the horizontal direction, the tension force is the only force we are concerned with.
#T = (m * v_T^2) /r #
If we then further complicate this, we can write an expression for a vertical circle, which is easiest at the top or bottom of the circle
#sum F = T - mg = (m * v_T^2) /r #
at the bottom of the circle and
#sum F = T + mg = (m * v_T^2) /r #
at the top of the circle.
More complicated examples include gravitation problems, where we estimate that a planet or the moon revolves around something in uniform circular motion. Here we would equate the gravitation force to the centripetal force
#(G m_1 m_2) / r_12^2 = (m * v_T^2) /r #
We can use this to derive Kepler's #T^2 prop r^3# law with a few substitutions based on our rotational kinematics rules.
Other more complex examples include loops in roller coasters (combining uniform circular motion with conservation laws) and other creative examples.
One example I saw dropped a pendulum that swung into a screw and entered uniform circular motion around the screw.
Creativity will allow a professor or instructor to create very difficult problems, but the above examples are the ones that a student in physics 1 should definitely see.