In inertial reference frame F0 there is a clock, C0, centered at (0,0). An identical clock, C1, is traveling in a circular path around that point with angular velocity V as measured in F0. When the traveling clock C1 makes a complete circle, which clock shows the greater elapsed time or do they show the identical elapsed times? (When the moving clock crosses the positive y axis of F0, set both clocks to zero. When the moving clock once again crosses the y-axis of F0, record the time of both clocks.)
As always, every time you use the word "when" to refer to simultaneous separate events, you need to specify the temporal foliation you are referring to. In terms of standard inertial coordinates x,y,t, in which C0 is stationary, and assuming that at t=t0 the readings of both clocks are set to zero and C1 crosses the positive y axis, and the next time C2 crosses the positive y axis is at t=t1, the reading of C0 at t=t1 will be t1, and the reading of C1 at t=t1 will be t1*sqrt(1-V^2/c^2).
Then during any arc length the traveling clock will show less time as you say then the
clock at the center of the circle.
Again, in terms of inertial coordinate system x,y,t in which C0 is stationary, the rate of elapsed proper time for C0 is dtau/dt = 1 and the rate of elapsed proper time for C1 is dtau/dt = sqrt(1-V^2/c^2).
During this arc length let another clock [C2] in an inertial reference frame travel in a straight
line with velocity V so that that clock and the clock traveling along the arc both have
almost identical velocities except the clock traveling on the arc travels in a very slightly
Yawn. This old canard. Are you, like, 10 years old?
Do both of these clocks run at nearly the same rate with the clock traveling
along the arc only very slightly slower?
Both clocks C1 and C2 run at the rate dtau/dt = sqrt(1-V^2/c^2). In general (flat spacetime), any clock moving at speed V in terms of inertial coordinates x,y,t runs at the rate dtau/dt = sqrt(1-V^2/c^2). Of course, in terms of inertial coordinates x',y',t' in which C2 is at rest (and C1 is momentarily at rest), clock C0 is moving at speed V, so it runs at the rate dtau/dt' = sqrt(1-V^2/c^2). A little while later, C1 will be at rest in a slightly different inertial coordinate system, x",y",t", and it will be running at the rate dtau/d" = 1, and clock C0 will be running at the rate dtau/dt" = sqrt(1-V^2/c^2). And so on.