124. Variations in the Force of Gravity at different Places of the Earth's Surface. - When experimental determinations of the force of gravity are made with great care, it is found to have different values at different places; these differences are due to two principal causes. (1) The spheroidal form of the earth, in consequence of which the attraction of the earth at different places is not the same. (2) The diurnal rotation of the earth, which causes the sensible or apparent force of gravity to be less than the actual attraction, as part of the latter is consumed in keeping bodies on the surface of the earth. Besides these general causes, variations are produced in the determinations made at particular places by differences in their level, and differences in the density of the strata in their immediate neighbourhood. The apparent force of gravity at any place is determined by ascertaining the length L of a simple pendulum which beats seconds at that place, and then the accelerative effect of gravity is determined by the formula (Ex. 646)

g=π2L

The preceding table gives the lengths of the seconds pendulum at different places, according to Mr. Airy,* and the values of g which can be deduced from them.

* Figure of the Earth, p. 229.

R

CHAPTER II.

ON UNIFORMLY ACCELERATED MOTION.

125. Accelerating Force. If the velocity of a body is continually increased by equal amounts in equal times, that velocity is said to be uniformly accelerated; and the cause which produces this acceleration is said to be a uniformly accelerating force.

Obs. If the velocity of a body is continually diminished by equal amounts in equal times, it is said to be uniformly retarded; and the cause which produces this effect is said to be a uniformly retarding force. Now, it must be remarked, the same cause may produce uniform acceleration in one body, and uniform retardation in another: thus gravity produces the former effect on a body moving downward, and the latter on a body moving upward: for this reason the term 'retarding force' is rarely used, and the accelerating force of gravity is spoken of, whether the body is moving upward or downward : some forces, however, such as friction, are essentially retarding forces. When a constant force acts on a body in the direction of its motion it produces a constant accelerative effect. Suppose that, measured in feet and seconds, this effect is f: this means that in each second of the body's motion its velocity is increased by a velocity of f feet per second. It is not unusual to speak of f as a uniformly accelerating force.

Proposition 23.

If ABCD be any area bounded by a line straight or curved CD, and by straight lines AB, AD, BC, of which the

two latter are at right angles to AB; and if the line DC be such that for any point P the ordinate PN represents the

velocity with which a body moves

at the end of a time t, that is represented on the same scale by AN, then the area of the curve will represent the space described by the body in the time a B.

Divide AB into any number of

...

D

A

P

P
2

FIG. 155.

3

P

4

P

C

5

N

N
2

N

N N

B

equal parts in N1, N2, N3, ... draw the ordinates P1N1, P2N29 P3N39. and complete the rectangles DN1, P1N2, P2N3,.... Now, if we suppose the body to move during each interval of time with the velocity it has at the commencement of that interval, it will (Art. 112) describe a space represented by the sum of the rectangular areas DN1, PIN2, P2N3, .....; and this will be true, however great the number of intervals may be, and therefore when the velocity changes continuously, the space described will be correctly represented by the limit of the sum of those areas, i. e. by the curvilinear area ABCD.

Proposition 24.

If a body begins to move along a straight line with a velocity of ▾ feet per second, and its motion undergoes a uniform acceleration f, the number of feet (s) described by it in t seconds is given by the formula

8=vt+ft2

Let A B represent the time t on scale; at right angles to A B draw A D and вс representing on the same scale v, the velocity at the beginning of the motion, and v+ft the velocity at the end of the motion (Art. 113); join DC,

PM:CE::DM: DE::AN: AB

and CE equals fXAB; therefore PM equals fx AN, therefore PN equals AD+fXAN, i. e. (Art. 113) represents the velocity with which the body is moving at the end of the time represented by AN; hence the area ABCD represents the required space s (Prop. 23).

Now area ABCD=AB(AD+BC)

If the velocity is uniformly retarded, a precisely similar

process leads to the equation

s=vt-ft2

and if the body begins to move from rest, we obtain

s=ft2.

Ex. 651. If a body is thrown up with any velocity, and if t1 and t2 are the times during which it is respectively above and below the middle point of its path, show that

t1:t2::1:2-1.

Ex. 652. If a body falls under the action of any uniformly accelerating force f, show that the spaces described in successive seconds form an arithmetical series of which the first term is and the common difference f.

f

2

Ex. 653. In the nth second of its motion a falling body describes h feet: determine its initial velocity. Ans. v=h-g(2n - 1).

Ex. 654. If a body is let fall and describes a certain space, and this space is divided into n equal parts, show that the time of describing the first part is to that of describing the last as 1 is to √ - √n-1.

Ex. 655. A falling body describes in the nth second m times the space described in the (n-1)th second; find the whole space described in the n seconds.

Ex. 656. There is a chasm with water at the bottom; on dropping a stone down it the splash is heard n seconds after the stone leaves the hand; show that the distance of the surface of the water below the hand is given by the formula (g=32.2)

s=1130 (35+n-1225 +70n)

[The velocity of sound may be taken at 1130 ft. per second; it will be observed also that 1130÷16.1 = 70 very nearly; now let a be the time the stone takes to fall, na is the time the sound takes to rise, and if s is the required depth we have

and

therefore

whence s is easily found.]

s=gx2=16.1 x2
s=1130 (n-x)

x2=70 (n-x)

Ex. 657. When n is but a few seconds, show that the formula in the last Example can be written

Ex. 658. Determine the values of s from the formulæ of Examples 656 and 657 when n equals 3, 4, and 5 seconds respectively.

Ans. (1) 134-3 ft. and 133.8 ft. (2) 232.4 ft. and 231.8 ft. (3) 354.5 ft. and 353.1 ft.

Ex. 659. A body during the 2nd, 5th, and 7th second of its motion describes respectively 16, 24, and 46 feet. Is this consistent with uniform acceleration?

Ex. 660. A body moves from rest under the action of a certain force; at the end of 5 sec. the force ceases to act, in the next 4 sec. the body describes 180 ft. Find the acceleration due to the force. Ans. 9.

126. Change in the numerical Value of an Accelerative Effect produced by a Change in the Units of Space and Time. We have hitherto taken the numerical value of the accelerative effect of gravity to be 32, which presupposes that space is measured in feet, and time in seconds; the choice of these units is of course arbitrary; the question then arises, were we to choose other units what numerical