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EXAMPLES. XIII.

1. A particle slides down an arc of a circle to the lowest point: find the velocity at the lowest point if the angle described round the centre is 60°.

2. If the length of the seconds' pendulum be 39.1393 inches find the value of g to three places of decimals.

3. A pendulum which oscillates in a second at one place is carried to another place where it makes 120 more oscillations in a day: compare the force of gravity at the latter place with that at the former.

4. Suppose that is the length of the seconds' pendulum, and that the lengths of two other pendulums are 1-c and l+c respectively, where c is very small: shew that the sum of the number of oscillations of these two pendulums in a day is very nearly 2 × 24 × 60 × 60 (1+ 812

3c2

5. A pendulum is found to make p oscillations at one place in the same time as it makes q oscillations at another. Shew that if a string hanging vertically can just support n cubic inches of a given substance at the former place it cubic inches at the latter place.

will just support

пра

6. A seconds' pendulum hangs against the smooth face of an inclined wall and swings in its plane: find the time of a small oscillation.

7. A seconds' pendulum is carried to the top of a mountain m miles high: assuming that the force of gravity varies inversely as the square of the distance from the centre of the earth, find the time of a small oscillation.

8. Shew that the length of a pendulum which will make a small oscillation in one second at the top of a moun2 4000 tain m miles high is

4000+

7,

„), where 7 is the length

m

of the seconds' pendulum at the surface of the earth.

XIV. Uniform motion in a Circle.

155. If the direction of a force always passes through a fixed point the force is called a central force; and the fixed point is called the centre of force.

Throughout the remainder of the present work we shall be occupied with cases of central forces: we begin with some propositions due to Newton which are contained in the next five Articles.

156. When a body moves under the action of a central force the areas described by the radius drawn to the centre of force are in one plane and are proportional to the times of describing them.

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be described by the radius drawn to S in equal intervals.

But when the body arrives at B let a force tending to S act on it by an impulse, and cause it to proceed in the direction BC instead of Bc; then if C be the position of the body at the end of the second interval Cc is parallel to BS: see Art. 47. Join SC; then the triangle BSC is equal to the triangle BSc, by Euclid, 1. 37; therefore the triangle BSC is equal to the triangle ASB, and the two triangles are in the same plane.

In like manner if impulses tending to Sact on the body at C, D, E,... causing the body to describe in successive equal intervals the straight lines CD, DE, ..., the triangles CSD, DSE,... are all equal to the triangle ASB, and are in the same plane.

Thus equal areas are described in equal intervals, and the sum of any number of areas is proportional to the time of description.

Now let the number of triangles be indefinitely increased, and the base of each indefinitely diminished; then the boundary ABCDE... will ultimately become a curve, and the series of impulses will become a continuous central force by the action of which the body is made to describe the curve. And the areas described being always proportional to the times will be so also in this case.

157. The proposition of the preceding Article is true also if S be a point which instead of being fixed moves uniformly in a straight line. For by Corollary 5 in Art. 142 the relative motion is the same whether the plane in which the curve is described be at rest or be moving with the body and the curve and the point S uniformly in a straight line.

158. If v be the velocity of the body at any point A, and p the perpendicular from S on the tangent at that point, the area described in the time t=ptv.

1

Draw SY perpendicular to AB. Let t be divided into n equal intervals, and let AB be the space described in the first interval, the force at S being supposed to act by impulses at the end of each interval.

Then the polygonal area which is described in the time t =n times the triangle SAB=nSY..v=SY.t.v.

n

2

In the limit the straight line AB, which is the direction of the velocity at A, becomes the tangent to the curve at A; and the curvilinear area described in the time

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Thus the area described in a unit of time is

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usual to denote twice the area described in a unit of

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159. If a body move in one plane so that the areas described by the radius drawn to a fixed point are proportional to the times of describing them the body is acted on by a force tending to that point.

Let S be the fixed point about which areas proportional to the times are described, and suppose a body acted on by no force to describe the straight line AB with uniform velocity in a given interval of time. In another equal interval if no force acted the body would describe Bc equal to AB, in AB produced: so that the triangles ASB and BSc would be equal. But when the body arrives at B let a force act on it by an impulse which causes it to describe BC in the second interval, such that the triangle SBC is equal to the triangle ASB, and in the same plane.

Then the triangle BSC is equal to the triangle BSc, and therefore Cc is parallel to SB, by Euclid, 1. 39: therefore the impulse at B is in the direction BS: see Art. 47.

In like manner if impulses act on the body at C, D, E,... causing the body to describe in successive equal intervals the straight lines CD, DE,... so that the triangles CSD, DSE,... are all equal to the triangle ASB, and are in the same plane, then all the impulses tend to S.

Hence if any polygonal areas be described proportional to the times of describing them, the impulses at the angular points all tend to S.

Now let the number of triangles be indefinitely increased, and the base of each indefinitely diminished; then the boundary ABCDE... will ultimately become a curve, and the series of impulses will become a continuous force by the action of which the body is made to describe the curve: and the force always tends to S.

160. The proposition of the preceding Article is true also if S be a point which instead of being fixed moves uniformly in a straight line; see Art. 157.

161. We have already observed in Art. 49 that the principle called the Parallelogram of Velocities gives rise to applications similar to those deduced from the Parallelogram of Forces in Statics; some illustrations of this remark will occur as we proceed, one of great interest being given in the next Article.

162. The direction of the resultant action of a central force on a body while it describes an arc of a curve is the straight line which joins the intersection of the tangents at the extremities of the arc with the centre of force.

Let PQ be an arc of a curve described by a body under the action of a centre of force at S. Let PT, QT be the tangents at P and Q respectively. Suppose the body to move from P to Q. Produce PT to any point p.

The resultant action of the central force during the motion changes the direction of the

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velocity from Tp to TQ; and thus the direction of the resultant action must pass through T. But the direction of the action of the central force passes through Sat every instant, and therefore the direction of the resultant action must pass through S. Thus TS must be the direction of the resultant action.

This proposition has been given on account of its simplicity and interest; but it is not absolutely necessary for the purposes of the present work, for it will be found that so much of the result as we may hereafter require will present itself naturally in the course of our investigations. See Arts. 163 and 175.

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