The answer is already posted at: http://www.enotes.com/science/q-and-a/this-true-false-all-objects-moving-circle-356384
The same is being reproduced below:
A very interesting question. Lets consider various scenarios to see what is happening.
a) Object moving in a circle with a constant angular velocity.
Since angular velocity is constant, there is no angular acceleration and the statement that object moving in a circle accelerates is False.
But it is not all. If we consiider this motion in a plane with respect to a stright line, the object is constantly changing its direction and velocity in with respect to any fixed direction. Hence it is undergoing certain amount of acceleration at all times. In this case the statement that object moving in a circle acceleratesis true.
b) Object moving in a circle with a variable angular velocity.
In this case as the we have stated that its angular velocity is changing so it is experiencing an angular acceleration and the statement that object moving in a circle accellerates is true. The same is true if we consider this motion with respect to any fixed direction.