Wakelin. — Fallacies in the Theory of Circular Motion. 135 



as to move in a circular orbit, of course round a central point, a certain force 

 must have acted on this body. Mr. Todhunter says: (1.) "If a body of 

 mass m describes a circle of radius r with uniform velocity v, then whatever 

 be the forces acting on the body their resultant tends to the centre of 

 the circle and is equal to — ^ . No single fact in the whole range 

 of dynamics is of greater importance than this." It will be advisable to 

 illustrate this measure of circular force by an example. A slinger whirls 

 round a stone in a sling. " The stone pulls at the string one way, the con- 

 trolling hand at the centre of its circle, the other (2). Were the string too 

 weak it would break, and the stone prematurely released would fly off in a 

 tangential direction. If a mechanist were told the weight of the stone (say 

 a pound), the length of the string (say a yard, including the motion of the 

 hand), and the number of turns made by the stone in a certain time (say 

 sixty in a minute or one in a second), he would be able to tell precisely what 

 ought to be the strength of the string so as just not to break : that is to say, 

 what weight it ought at least to be able to lift without breaking. In the 

 case I have mentioned it ought to be capable of sustaining 3 lbs. 10 oz. 

 386 grs. If it be weaker it will break. And this is the force or effort which 

 the hand must steadily exert to draw the stone in towards itself, out of the 

 direction in which it would naturally proceed if let go, and to keep it re- 

 volving in a circle at that distance." The result of the foregoing example is 

 obtained from the preceding formula in this way. The formula stated that 

 the acceleration, that is the pull on the hand of the slinger, is equal to the 

 mass (one pound) multiplied by the square of the velocity divided by the 

 radius. As the radius was three feet, the circumference of the circle would 

 be equal to 18-8496 feet nearly, and as the mass made one revolution in 

 a second, this is the velocity. The square of this velocity is equal to 

 355-30742016 feet, and on dividing this by the radius, three feet, the accele- 

 ration is found to be equal to 118-43580672 feet. The acceleration of the 

 force of gravity at the surface of the earth is very nearly equal to 32-2 feet, 

 and it is thus seen that the force with which the stone has to be pulled into 

 the hand greatly exceeds the force with which a pound weight is pulled to 

 the earth by gravity. On dividing 118* etc., by 32-2 the answer is 3-6781306, 

 which is the ratio of the greater acceleration to the less — that is, the force 

 said to be pulling the stone (a pound mass) into the hand is 3-67, etc., times 

 the force with which gravity pulls a pound mass towards the earth. Now 

 the force of gravity pulling at the stone gave it a weight equal to one pound, 

 and consequently the central force pulling at the stone would give it a weight 

 of 3-6781306 pounds, meaning that the hand has to bear this weigh*; to 

 keep the stone from breaking away. This reduced is equal to 3 lbs. 10 oz. 

 371-9142 grains, and more exact figures would have given a closer result. 



