A detailed analysis is presented for the problem of uniform circular motion in which the centripetal force is provided by a stretched spring whose mass is not negligible compared with that of the whirling body. It is shown that the force required to stretch the spring statically to the length it has during the rotation exceeds the centripetal force it exerts on the whirling body by an amount depending in a simple way on the spring mass; the excess can be erased by adding to the body mass approximately one-third the spring mass in computing the centripetal force. Some experimental results that tend to support the theory are exhibited and discussed. A computation of radial-oscillation frequencies is included.