Proposition 7 Problem 2

The law of centripetal force directed to any given point in a body revolving in a circle

Let:

• VQPA be the circumference of the circle
• S is the given point which is the center where the force tends
• P is the body moving in the
• Q is the next place into which it is to move
• PRZ is the tangent of the circle at the preceding place

Through point S, draw the chord PV, and the diameter VA of the circle. Join AP and draw QT perpendicular to SP which produced may meet the tangent PR in Z.

Lastly, through the point Q, draw LR parallel to SP, meeting the circle in L, and the tangent PZ in R. And, because of the similar triangles ZQR, ZTP, VPA, we shall have RP², that is, QRL to QT² as AV² to PV².

Therefore, … is equal to QT².

Multiply those equals by SP2/QR, and the points P and Q coinciding, for RL write PV; then we shall have

…

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

…

That is (because AV² is given), reciprocally as the square of the distance or altitude SP, and the cube of the chord PV conjunctly. Q.E.I.

The same otherwise.

On the tangent PR produced let fall the perpendicular SY; and (because of the similar triangles SYP, VPA), we shall have AV to PV as SP to SY, and therefore

…

and

…

Therefore (by Corol. 3 and 5, Prop. VI), the centripetal force is reciprocally as

…

that is (because AV is given), reciprocally as

…

Q.E.I.

Cor. 1

Hence if the given point S, to which the centripetal force always tends, is placed in the circumference of the circle, as at V, the centripetal force will be reciprocally as the quadrato-cube (or fifth power) of the altitude SP.

Cor. 2

The force by which the body P in the circle APTV revolves about the centre of force S is to the force by which the same body P may revolve in the same circle, and in the same periodic time, about any other centre of force R, as

{\displaystyle \scriptstyle RP^{2}\times SP} to the cube of the right line SG, which, from the first centre of force S is drawn parallel to the distance PR of the body from the second centre of force R, meeting the tangent PG of the orbit in G. For by the construction of this Proposition, the former force is to the latter as

…

to

that is, as

to

or (because of the similar triangles PSG, TPV) to SG³.

Cor. 3

The force by which the body P in any orbit revolves about the centre of force S, is to the force by which the same body may revolve in the same orbit, and the same periodic time, about any other centre of force R, as the solid

contained under the distance of the body from the first centre of force S, and the square of its distance from the second centre of force R, to the cube of the right line SG, drawn from the first centre of the force S, parallel to the distance RP of the body from the second centre of force R, meeting the tangent PG of the orbit in G. For the force in this orbit at any point P is the same as in a circle of the same curvature.