Question

If an object moves in uniform circular motion in a circle of radius R = 1.0 meter, and the object takes 4.0 seconds to complete ten revolutions, calculate the magnitude of the velocity around the circle. (Note: Remember, 10 revolutions is a counting number and not a measurement.)

Answer 1

Radius = 1.0 meter
Diameter = 2.0 meters
Circumference = 2 π meters
10 revolutions = 20 π meters

Speed = (10 revs) / (4 sec)

= (20 π meters) / (4 seconds)

= 5 π m/s

= 15.708 m/s (rounded)

Answer 2

Let's supose that your object is the orange rectangle rotating around the blue circle (see picture attached)

r = 1.0 [m] is the radius of the circle (it's R in your problem).
ω [rad/s] - is the angular velocity of the object, it is measured in radians per second. We will compute it from the data you have
v [m/s] - is the tangential/linear speed, it is measured in meters per second

We know that the rotational speed is 10 revolutions per 4 seconds.

10/4 rev/s = 2.5 rev/s

We convert rev/s in rad/s and we get:
2.5 rev/s = 15.708 rad/s

So we found our angular speed: ω = 15.708 rad/s

The relationship between v and ω is:

v = ω · r = 15.708 · 1 = 15.708 m/s

Answer: the magnitude of the velocity around the circle is 15.708 m/s