PROBLEM 7

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 in and lastly, through PR Z the point Q, draw tangent PZ in R.

LR parallel to SP, RP VPA, we shall therefore QRlj x PV

have 2 that , TS – is equal and the points SP- X PV 5 meeting the circle" in L, and the ZQR, ZTP. AV to PV And And, because of the similar triangles is. to P and Q, coinciding, SP x QT 2 QRL

Multiply those equals by RL write PV therefore flr Cor for ;

. then we shall have 2 And 1 and 5. Prop. VI.)

[BOOK SP 2 X PV 3 the centripetal force is reciprocally as - ia ry^~ J that is (because given), reciprocally as the square of the distance or altitude 3ube of the chord PV conjunctly. otherwise. PR SY, and And therefore 2 SP, and the SY and (be AV to PV as SP the tangent produced let fall the perpendicular cause of the similar triangles SYP, VPA), we shall have to AV Q.E.L The same On I, SP X PV -

= SY, and - SP ^~ A V PV ^- 3 2 >< A V

; SY 2 X PV. therefore (by Corol. 3 and 5, Prop. VI), the centripetal force is recip- rocally as X PV 3 ry~ I

• na *
• s (because AV is given), reciprocally as SP" Q.E.I. . Hence if the given point S, to which the centripetal force al tends, is placed in the circumference of the circle, as at V, the cen Con. ways 2 SP X PV 3

Corollary 2

tripetal force will be reciprocally as the quadrato-cube (or fifth power) of the altitude COR. circle 2. SP. The APTV force by which the body P in the revolves about the centre of force S by which the same body P may re volve in the same circle, and in the same periodic is to the force 2 any other centre of force R, as RP X the cube of the right line SG, which, from time, about SP to 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, that as the former force is to the latter as RP X PT to SP X PV 2 3 2 3 ; SP X PV

3 SP X RP to SG S 2 to p is, 3 ; or (because of the similar triangles PSG, TPV) . by which the body P in any orbit revolves about the the force by which the same body may revolve in the same orbit, and the same periodic time, about any other centre of force 2 R. as the solid SP X RP contained under the distance of the body from the first centre of force S, and the square of its distance from the sec ond 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 COR. 3. The force centre of force S, is to ,

fch*3 PG of the orbit in G. second centre of force R, meeting the tangent this orbit at any point P is the same as in a circle of the For the force in same curvature.SJSG.