Proposition 8 Problem 3

Proposition 8 Problem 3

If a body moves in the semi-circumference PQA,find the law of the centripetal force tending to a point S, so remote that all the lines PS, RS drawn thereto may be taken for parallels.

C is the centre of the semi-circle. From it, draw the semi-diameter CA.

It cuts the parallels at right angles in M and N, and join CP.

Because of the similar triangles CPM, PZT, and RZQ, we shall have CP² to PM² as PR² to QT²; and, from the nature of the circle, PR² is equal to the rectangle QR x RN x QN or, the points P, Q, coinciding, to the rectangle QR x 2PM.

Therefore CP² is to PM² as to QT²; and

…

and

….

Therefore (by Corol. 1 and 5; Prop. VI.), the centripetal force is reciprocally as

…

that is (neglecting the given ratio

…

reciprocally as PM³. Q.E.I.

The same thing is likewise easily inferred from the preceding Proposition.

SCHOLIUM

A body will be moved in an ellipsis, or even in an hyperbola, or parabola, by a centripetal force which is reciprocally as the cube of the ordinate directed to an infinitely remote centre of force.