The arc length traveled in a time ∆t by an object going in a circular orbit with uniform speed is the geometric mean of the circle’s diameter and the distance that the object would fall if it experienced a constant acceleration whose magnitude is equal to the magnitude of the centripetal force.
using this fact,
a.) draw a diagram that shows a circular orbit, a small segment of the orbit that the object traverses in a time ∆t, the arc length associated with that segment of the orbit, and the direction of the centripetal force.
b.) prove the proposition: derive the claimed relationship between arc length, diameter, and the distance that a particle would fall under the influence of a constant force with the same magnitude as the centripetal force.
tip: if the speed is constant then the arc length traversed in a time ∆t can be easily computed, as can the area swept out, and there will be a simple relationship between the speed and the magnitude of the centripetal force.