PENDULUM

PENDULUM, a vibrating body suspended from a fixed point. For the history of this invention, see the article CLOCK.

The theory of the pendulum depends on that of the inclined plane. Hence, in order to understand the nature of the pendulum, it will be necessary to premise some of the properties of this plane; referring, however to Inclined PLANE, and to the article MECHANICS, for the demonstration.

I. Let AC (fig. 1.) be an inclined plane, AB its perpendicular height, and D any heavy body: then the force which impels the body D to descend along the inclined plane AC, is to the absolute force of gravity as the height of the plane AB is to its length AC; and the motion of the body will be uniformly accelerated.

II. The velocity acquired in any given time by a body descending on an inclined plane AC, is to the velocity acquired in the same time by a body falling freely and perpendicularly, as the height of the plane AB to its length AC. The final velocities will be the same; the spaces described will be in the same ratio; and the times of description are as the spaces described.

III. If a body descend along several contiguous planes,

Pendulum planes, AB, BC, CD, (fig. 2.) the final velocity, namely, that at the point D, will be equal to the final velocity in descending through the perpendicular AE, the perpendicular heights being equal. Hence, if these planes be supposed indefinitely short and numerous, they may be conceived to form a curve, and therefore the final velocity acquired by a body in descending through any curve AF, will be equal to the final velocity acquired in descending through the planes AB, BC, CD, or to that in descending through AE, the perpendicular heights being equal.

IV. If from the upper or lower extremity of the vertical diameter of a circle a cord be drawn, the time of descent along this cord will be equal to the time of descent through the vertical diameter; and therefore the times of descent through all cords in the same circle, drawn from the extremity of the vertical diameter, will be equal.

V. The times of descent of two bodies through two planes equally elevated will be in the subduplicate ratio of the lengths of the planes. If, instead of one plane, each be composed of several contiguous planes similarly placed, the times of descent along these planes will be in the same ratio. Hence, also, the times of describing similar arches of circles similarly placed will be in the subduplicate ratio of the lengths of the arches.

VI. The same things hold good with regard to bodies projected upward, whether they ascend upon inclined planes or along the arches of circles.

The point or axis of suspension of a pendulum is that point about which it performs its vibrations, or from which it is suspended.

The centre of oscillation is a point in which, if all the matter in a pendulum were collected, any force applied at this centre would generate the same angular velocity in a given time as the same force when applied at the centre of gravity.

The length of a pendulum is equal to the distance between the axis of suspension and centre of oscillation.

Fig. 3. Let PN (fig. 3.) represent a pendulum suspended from the point P; if the lower part N of the pendulum be raised to A, and let fall, it will by its own gravity descend through the circular arch AN, and will have acquired the same velocity at the point N that a body would acquire in falling perpendicularly from C to F, and will endeavour to go off with that velocity in the tangent ND; but being prevented by the rod or cord, will move through the arch NB to B, where, losing all its velocity, it will by its gravity descend through the arch BN, and, having acquired the same velocity as before, will ascend to A. In this manner it will continue its motion forward and backward along the arch ANB, which is called an oscillatory or vibratory motion; and each swing is called a vibration.

PROP. I. If a pendulum vibrates in very small circular arches, the times of vibration may be considered as equal, whatever be the proportion of the arches.

Fig. 4. Let PN (fig. 4.) be a pendulum; the time of describing the arch AB will be equal to the time of describing CD; these arches being supposed very small.

Join AN, CN; then since the times of descent along all cords in the same circles, drawn from one extremity of the vertical diameter, are equal; therefore the cords AN, CN, and consequently their doubles, will be described in the same time; but the arches AN, CN being supposed very small, will therefore be nearly equal to their cords; hence the times of vibrations in these arches will be nearly equal.

PROP. II. Pendulums which are of the same length vibrate in the same time, whatever be the proportion of their weights.

This follows from the property of gravity, which is always proportional to the quantity of matter, or to its inertia. When the vibrations of pendulums are compared, it is always understood that the pendulums describe either similar finite arcs, or arcs of evanescent magnitude, unless the contrary is mentioned.

PROP. III. If a pendulum vibrates in the small arc of a circle, the time of one vibration is to the time of a body's falling perpendicularly through half the length of the pendulum, as the circumference of a circle is to its diameter.

Let PE (fig. 5.) be the pendulum which describes the arch ANC in the time of one vibration; let PN be perpendicular to the horizon, and draw the cords AC, AN; take the arc Ee infinitely small, and draw EFG, efg perpendicular to PN, or parallel to AC; describe the semicircle BGN, and draw er, gs perpendicular to EG; now let t = time of descending through the diameter 2PN, or through the cord AN: Then the velocities gained by falling through 2PN, and by the pendulum's descending through the arch AE, will be as $\backslash sqrt\{2PN\}$ and $\backslash sqrt\{BF\}$; and the space described in the time t, after the fall through 2PN, is 4PN. But the times are as the spaces divided by the velocities.

Therefore $\backslash frac\{4PN\}\{\backslash sqrt\{2PN\}\}$ or $2\backslash sqrt\{2PN\}\; :\; t\; ::\; \backslash frac\{Ee\}\{\backslash sqrt\{BF\}\}$ : time of describing Ee = $\backslash frac\{t\; \backslash times\; Ee\}\{2\backslash sqrt\{2PN\}\; \backslash times\; BF\}$. But in the similar triangles PEF, Eer, and KGF, $Ggs$,

As $PE\; =\; PN\; :\; EF\; ::\; Ee\; :\; er\; =\; \backslash frac\{EF\}\{PN\}\; \backslash times\; Ee$,
And $KG\; =\; KD\; :\; FG\; ::\; Gg\; :\; gs\; =\; \backslash frac\{FG\}\{KD\}\; \backslash times\; Gg$.

But $er\; =\; Ggs$; therefore $\backslash frac\{EF\}\{PN\}\; \backslash times\; Ee\; =\; \backslash frac\{FG\}\{KD\}\; \backslash times\; Gg$.

Hence $Ee\; =\; \backslash frac\{PN\; \backslash times\; FG\}\{KD\; \backslash times\; EF\}\; \backslash times\; Gg$.

And by substituting this value of Ee in the former equation, we have the time of describing Ee = $\backslash frac\{t\; \backslash times\; PN\; \backslash times\; FG\; \backslash times\; Gg\}\{2KD\; \backslash times\; KF\; \backslash times\; \backslash sqrt\{BF\}\; \backslash times\; 2PN\}$. But by the nature of the circle $FG\; =\; \backslash sqrt\{BF\; \backslash times\; FN\}$, and $EF\; =\; \backslash sqrt\{PN\; +\; PF\; +\; FN\}$.

Hence, by substitution, we obtain the time of describing Ee = $\backslash frac\{t\; \backslash times\; PN\; \backslash times\; \backslash sqrt\{BF\; \backslash times\; FN\}\; \backslash times\; Gg\}\{2KD\; \backslash times\; \backslash sqrt\{PN\; +\; PF\}\; \backslash times\; \backslash sqrt\{PN\; +\; PF\}\; \backslash times\; \backslash sqrt\{BF\}\; \backslash times\; 2PN\}\; =\; \backslash frac\{t\; \backslash times\; \backslash sqrt\{PN\}\; \backslash times\; Gg\}\{2KD\; \backslash times\; \backslash sqrt\{PN\; +\; PF\}\; \backslash times\; \backslash sqrt\{2\}\}\; =\; \backslash frac\{t\; \backslash times\; \backslash sqrt\{2PN\}\; \backslash times\; Gg\}\{4KD\; \backslash times\; \backslash sqrt\{PN\; +\; PF\}\}\; =\; \backslash frac\{t\; \backslash times\; \backslash sqrt\{2PN\}\}\{2BN\; \backslash times\; \backslash sqrt\{2PN\; -\; NF\}\}\; \backslash times\; Gg$. But NF, in its mean quantity for all the arches Gg, is nearly equal to NK; for if the semicircle described on the diameter BN, which corresponds to the whole arch AN, be divided into

Pendulum. into an indefinite number of equal arches, $Gg$, &c. the sum of all the lines $NF$, will be equal to as many times $NK$, as there are arches in the semicircle equal to $Gg$;

therefore the time of describing $Ee\; =\; \backslash frac\{t\; \backslash times\; \backslash sqrt\{2PN\}\}\{2BN\; \backslash times\; \backslash sqrt\{2PN\}\; -\; NK\}$

$\backslash times\; Gg$. Whence the time of describing the arch $AED$

$=\; \backslash frac\{t\; \backslash times\; \backslash sqrt\{2PN\}\}\{2BN\; \backslash times\; \backslash sqrt\{2PN\}\; -\; NK\}\; \backslash times\; BGN$; and the time of describing the whole arch $ADC$, or the time of one vibration, is $=\; \backslash frac\{t\; \backslash times\; \backslash sqrt\{2PN\}\}\{2BN\; \backslash times\; \backslash sqrt\{2PN\}\; -\; NK\}\; \backslash times\; 2BGN$. But

when the arch $ANC$ is very small, $NK$ vanishes, and then the time of vibration in a very small arc is

$=\; \backslash frac\{t\; \backslash times\; \backslash sqrt\{2PN\}\}\{2BN\; \backslash times\; \backslash sqrt\{2PN\}\}\; \backslash times\; 2BGN\; =\; \backslash frac\{t\}\{BN\}\; \backslash times\; \backslash frac\{2BGN\}\{BN\}$. Now if $t$

be the time of descent through $2PN$; then since the spaces described are as the squares of the times, $\backslash frac\{1\}\{2\}t$ will be the time of descent through $\backslash frac\{1\}\{2\}PN$: therefore the diameter $BN$ is to the circumference $2BGN$, as the time of falling through half the length of the pendulum is to the time of one vibration.

PROP. IV. The length of a pendulum vibrating seconds is to twice the space through which a body falls in one second, as the square of the diameter of a circle is to the square of its circumference.

Let $d\; =$ diameter of a circle $=\; 1$, $c\; =$ circumference $=\; 3.14159$, &c. $t$ to the time of one vibration, and $p$ the length of the corresponding pendulum; then by the

last proposition $c\; :\; d\; ::\; 1\text{'}\text{'}\; :\; \backslash frac\{d\}\{c\}\; =$ time of falling through half the length of the pendulum. Let $s\; =$ space described by a body falling perpendicularly in the first second: then since the spaces described are in the subduplicate ratio of the times of description, therefore

$1\text{'}\text{'}\; :\; \backslash frac\{d\}\{c\}\; ::\; \backslash sqrt\{s\}\; :\; \backslash sqrt\{\backslash frac\{1\}\{2\}p\}$. Hence $c^2\; :\; d^2\; ::\; 2s\; :\; p$.

It has been found by experiment, that in latitude $51^\{\backslash circ\}40\text{'}$ a body falls about 16.11 feet in the first second: hence the length of a pendulum vibrating seconds in

that latitude is $=\; \backslash frac\{32.22\}\{3.14159^2\}\; =\; 3$ feet 3.174 inches.

PROP. V. The times of the vibrations of two pendulums in similar arcs of circles are in a subduplicate ratio of the lengths of the pendulums.

Let $PN,\; PO$ (fig. 6.) be two pendulums vibrating in the similar arcs $AB,\; CD$; the time of a vibration of the pendulum $PN$ is to the time of a vibration of the pendulum $PO$ in a subduplicate ratio of $PN$ to $PO$.

Since the arcs $AN,\; CO$ are similar and similarly placed, the time of descent through $AN$ will be to the time of descent through $CO$ in the subduplicate ratio of $AN$ to $CO$: but the times of descent through the arcs $AN$ and $CO$ are equal to half the times of vibration of the pendulums $PN,\; PO$ respectively. Hence the time of vibration of the pendulum $PN$ in the arc $AB$ is to the time of vibration of the pendulum $PO$ in the similar arc $CD$ in the subduplicate ratio of $AN$ to $CO$: and since the radii $PN,\; PO$ are proportional to the similar arcs $AN,\; CO$, therefore the time of vibration of the pendulum $PN$ will be to

the time of vibration of the pendulum $PO$ in a subduplicate ratio of $PN$ to $PO$.

If the length of a pendulum vibrating seconds be 39.174 inches, then the length of a pendulum vibrating half seconds will be 9.793 inches. For $1\text{'}\text{'}\; :\; \backslash frac\{1\}\{2\}\text{'}\text{'}\; ::\; \backslash sqrt\{39.174\}\; :\; \backslash sqrt\{x\}$; and $1\; :\; \backslash frac\{1\}\{2\}\; ::\; 39.174\; :\; x$. Hence $x\; =\; \backslash frac\{39.174\}\{4\}\; =\; 9.793$.

PROP. VI. The lengths of pendulums vibrating in the same time, in different places, will be as the forces of gravity.

For the velocity generated in any given time is directly as the force of gravity, and inversely as the quantity of matter. Now the matter being supposed the same in both pendulums, the velocity is as the force of gravity; and the space passed through in a given time will be as the velocity; that is, as the gravity.

Cor. Since the lengths of pendulums vibrating in the same time in small arcs are as the gravitating forces, and as gravity increases with the latitude on account of the spheroidal figure of the earth and its rotation about its axis; hence the length of a pendulum vibrating in a given time will be variable with the latitude, and the same pendulum will vibrate slower the nearer it is carried to the equator.

PROP. VII. The time of vibrations of pendulums of the same length, acted upon by different forces of gravity, are reciprocally as the square roots of the forces.

For when the matter is given, the velocity is as the force and time; and the space described by any given force is as the force and square of the time. Hence the lengths of pendulums are as the forces and the squares of the times of falling through them. But these times are in a given ratio to the times of vibration; whence the lengths of pendulums are as the forces and the squares of the times of vibration. Therefore, when the lengths are given, the forces will be reciprocally as the square of the times, and the times of vibration reciprocally as the square roots of the forces.

Cor. Let $p\; =$ length of pendulum, $g\; =$ force of gravity, and $t\; =$ time of vibration. Then since $p\; =\; g\; \backslash times\; t^2$. Hence $g\; =\; p\; \backslash times\; \backslash frac\{1\}\{t^2\}$; and $t\; =\; \backslash sqrt\{p\; \backslash times\; \backslash frac\{1\}\{g\}\}$.

That is, the forces in different places are directly as the lengths of the pendulums, and inversely as the square roots of the times of vibration; and the times of vibration are directly as the square roots of the lengths of the pendulums, and inversely as the square roots of the gravitating forces.

PROP. VIII. A pendulum which vibrates in the arch of a cycloid describes the greatest and least vibrations in the same time.

This property is demonstrated only on a supposition that the whole mass of the pendulum is concentrated in a point: but this cannot take place in any really vibrating body; and when the pendulum is of finite magnitude, there is no point given in position which determines the length of the pendulum; on the contrary the centre of oscillation will not occupy the same place in the given body, when describing different parts of the tract it moves through, but will continually be moved in respect of the pendulum itself during its vibration. It may, however, be observed, that Huyghens, aware that

Pendulum a pendulum ball suspended at the end of a thread vibrating between cycloids, would not describe a cycloid with its center of oscillation, gave a very beautiful and simple method of suspension, which secured its vibrations in that curve. Harrison, whose authority is next, insists on the advantage of wide vibrations, and in his own clocks, he always used cycloidal cheeks. This circumstance has prevented any general determination of the time of vibration in a cycloidal arc, except in the imaginary case referred to.

There are many other obstacles which concur in rendering the application of this curve to the vibration of pendulums designed for the measures of time the source of errors far greater than those which by its peculiar property it is intended to obviate; and it is now wholly disused in practice.

Although the times of vibration of a pendulum in different arches be nearly equal, yet from what has been said, it will appear, that if the ratio of the least of these arches to the greatest be considerable, the vibrations will be formed in different times; and the difference, though small, will become sensible in the course of one or more days. In clocks used for astronomical purposes, it will therefore be necessary to observe the arc of vibration; which if different from that described by the pendulum when the clock keeps time, there a correction must be applied to the time shown by the clock. This correction, expressed in seconds of time, will be equal to the half of three times the difference of the square of the given arc, and of that of the arc described by the pendulum when the clock keeps time, these arcs being expressed in degrees; and so much will the clock gain or lose according as the first of these arches is less or greater than the second.

Thus, if a clock keep time when the pendulum vibrates in the arc of $3^\{\backslash circ\}$, it will lose $10\backslash frac\{1\}\{2\}$ seconds daily in an arc of $4^\{\backslash circ\}$.

For $4^3\; -\; 3^3\; =\; 7\; \backslash times\; \backslash frac\{1\}\{2\}\; =\; 10\backslash frac\{1\}\{2\}$ seconds.

The length of a pendulum rod increases with heat; and the quantity of expansion answering to any given degree of heat is experimentally found by means of a pyrometer; but the degree of heat at any given time is shown by a thermometer: hence that instrument should be placed within the clock case at a height nearly equal to that of the middle of the pendulum; and its height, for this purpose, should be examined at least once a day. Now by a table constructed to exhibit the daily quantity of acceleration or retardation of the clock answering to every probable height of the thermometer, the corresponding correction may be obtained. It is also necessary to observe, that the mean height of the thermometer during the interval ought to be used. In Six's thermometer this height may be easily obtained; but in thermometers of the common construction it will be more difficult to find this mean.

It had been found, by repeated experiments, that a brass rod equal in length to a second pendulum will expand or contract $\backslash frac\{1\}\{250\}$ part of an inch by a change of temperature of one degree in Fahrenheit's thermometer; and since the times of vibration are in a subduplicate ratio of the lengths of the pendulum, hence an expansion or contraction of $\backslash frac\{1\}\{250\}$ part of an inch will answer nearly to one second daily: therefore a change of one degree in the thermometer will occasion a difference in the rate of the clock equal to one second daily.

VOL. XVI. Part I.

When the clock be so adjusted as to keep time Pendulum when the thermometer is at $55^\{\backslash circ\}$, it will lose 10 seconds daily when the thermometer is at $65^\{\backslash circ\}$, and gain as much when it is at $45^\{\backslash circ\}$.

Hence the daily variation of the rate of the clock from summer to winter will be very considerable. It is true indeed that most pendulums have a nut or regulator at the lower end, by which the bob may be raised or lowered a determinate quantity; and therefore, while the height of the thermometer is the same, the rate of the clock will be uniform. But since the state of the weather is ever variable, and as it is impossible to be raising or lowering the bob of the pendulum at every change of the thermometer, therefore the correction formerly mentioned is to be applied. This correction, however, is in some measure liable to a small degree of uncertainty; and in order to avoid it altogether, several contrivances have been proposed by constructing a pendulum of different materials, and so disposing them that their effects may be in opposite directions, and thereby counterbalance each other; and by this means the pendulum will continue of the same length.