By twirling either the single ball on the string or the two balls on the string, you can demonstrate centripetal acceleration in two ways:
By swinging the ball(s) in a horizontal plane and then releasing it (them), you can show that the ball(s) would fly off along a straight line if it (they) could – tangent to its (their) circular path at the point of release, but that the string provides a centripetal acceleration, which keeps it (them) moving around in a circle. Expressed in terms of the ball’s (or balls’) instantaneous tangential speed, v, and the radius of the circle, r, this acceleration is v2/r. (See demonstration 16.03 -- Ball on turntable.) The centripetal force, then, which is the tension in the string, is mv2/r, where m is the mass of the ball. For the string with two balls, the two parts of the string have different tensions, since the masses are not both at the same point.
When you swing the ball(s) in a vertical plane, then at the bottom of the circle, the centripetal force provided by the string opposes gravity, and at the top of the circle, the centripetal force acts in the same direction as gravity. Thus, if you twirl the ball(s) at such a speed that at the top of the swing gravity provides the centripetal acceleration necessary to keep the ball(s) moving in a circle, the tension in the string at this point is zero, and below this speed gravity causes the ball(s) to take a flattened path at the top. That is, it (they) falls (fall) below the circular path it (they) would take if the centripetal acceleration needed to keep it (them) in that path equaled or exceeded that due to gravity. With the double-ball arrangement, since for any rate of revolution each ball has the same v/r, but the acceleration goes as v2/r, it is not possible to demonstrate the aforementioned deviation from circular motion first with one ball and then the other, since the inner ball’s orbit should flatten before that of the outer one.
It is possible, however, to have fun with the two-ball arrangement in at least one other way. The inner tennis ball is held in place by friction; that is, you can slide it along the string if you pull hard enough. If you twirl the balls fast enough around your head, the centripetal acceleration necessary to hold the ball in place, mv2/r, exceeds the friction, and the ball slides outward until it stops against the outer ball.