G. P. (George Payn) Quackenbos.

A natural philosphy: embracing the most recent discoveries in the various branches of physics .. online

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as loug in swinging from E to F as from C to D. The shorter the arc, there-
fore, the slower its motion. It is on this principle that a swing, when first
set in motion, goes very slowly, but increases in velocity as it is pushed
higher and higher.

141. Second Law. TJie vibrations of pendulums of
different length are performed in different times ; and
their lengths are proportioned to the squares of their times
of vibration.

One pendulum vibrates in 2 seconds, another in 4. Then the latter will
be four times as long as the former ; because they will be to each other as
the square of 2 is to the square of 4, that is, as 4 is to 16. Hence, to have
its time of vibration doubled, a pendulum must be made 4 times as long ; to
have it tripled, 9 times as long ; to have it quadrupled, 1G times as long, &c.
A pendulum, to vibrate only once in a minute, would have to be 60 times CO,
that is 3,600, times as long as one that vibrates once in a second, or a little
over 2 miles.

Conversely, the times in which different pendulums vibrate are to each
other as the square roots of their length. If one pendulum be 16 feet loug
and another 4, the former will be twice as long in vibrating as the latter;
for their times of vibration are to each other as the square root of 16 is to
the square root of 4, or as 4 to 2.

142. Third Laic. The vibrations of the same pendu-
lum are not performed in, the same time at all parts of the
earth's surface ; but, being caused by gravity, differ slight-
ly, like gravity, according to the distance from the earths
centre.

On the top of a mountain five miles high, for instance, a pendulum vibrat-
ing seconds would make 10 less vibrations in an hour than at the level of the
sea, because it would be farther from the earth's centre. At either pole, a
second-pendulum would make 13 more vibrations in an hour than at Ihe equa-
tor, because it is nearer the centre, the earth being flattened at the poles.
Hence the vibrations of the pendulum afford a means of measuring heights.

140. "What is the first law relating to the pendulum ? Illustrate this with Fte. 56.
111. "What is the second law? Apply this law in $n example. When the lengths of
different pendulums are known, how can we find the relative times of vibration ? If
we have two pend'ilnms. 16 and 4 feet Ion?, how will their times of vihration com-
pare? 142. "What is the third law? What is the difference in the number of vibra-
tions in a second-pendulum at the level of th sea an 1 at an elevation of five miles ?
flow would the number of vibrations at the pole compare with those at the equator ?



THE PENDULUM APPLIED TO CLOCK-WOKK. 6V

They also confirm what we have learned, that the polar diameter of the earth
is 26 4 miles shorter than its equatorial diameter.

In the latitude of New York, a pendulum, to vibrate seconds, must be
about SI) 1 / inches long; whereas at Spitzbergen, in the far North, it must
be a little over 3y Ya> an( l a ^ the equator exactly 30 inches.

143. APPLICATION OF THE PENTJULUM TO CLOCK-WOKK.
Galileo, to whom science owes so much, was the first to
think of turning the pendulum to a practical use. Observ-
ing that a chandelier suspended from the ceiling of a church
in Pisa, when moved by the wind, vibrated in exactly the
same time, whether carried to a greater or less distance, he
at once saw that a similar instrument might be employed
in measuring small intervals of time in astronomical obser-
vations.

To adapt it to this use, it was necessary to invent some
way of counterbalancing the constant loss of motion caused
by friction and the air's resistance. This was done by the
Dutch astronomer Huygens \]ii'-genz\, who in the year
1656 first applied the pendulum to clock-work. To this
great invention modern astronomy owes its precision of ob-
servation, and consequently much of the progress it has
made.

144. As a pendulum vibrating seconds, which is over
39 inches long, would be inconvenient in clocks, it is custom-
ary to use one that vibrates half-seconds ; which, according
to the principles laid down in 141, is one-fourth as long,
or a little less than 10 inches.

145. At the same distance from the equator, the same
elevation above the sea, and the same temperature, a pen-
dulum of given length will always vibrate in exactly the
same time, and a clock regulated by a pendulum will
keep uniform time. If taken from the equator towards the
poles, the pendulum will vibrate more rapidly, and the clock

What is the length of a second-pendulum at New York ? At Spitzbergen? At the
equator? 143. Who first thought of turning the pendulum to a practical use ? He-
late the circumstance that led him to do so. To enable it to measure small intervals
Df time, what was first necessary ? Who did this, and thus first applied the pendu-
lum to clock-work ? 144. What is the length of the pendulums generally used in
clocks ? 145. Under what circumstances will a pendulum always vibrate iu the same



68 MECHANICS.

will go too fast. If taken up a mountain, the pendulum
will vibrate less rapidly, and the clock will go too slow. If
expanded by the heat of summer (for such we shall here-
after learn is the effect of heat), the pendulum will also vi-
brate less rapidly, and the clock will go too slow.

146. THE GRIDIRON PENDULUM. To prevent a clock
from being affected by heat and cold, the Compensation
Pendulum is used.

Fig. 55. One form of the Compensation Pendulum, known as the Grid-

iron Pendulum, is represented in Fig. 55. It consists of a frame
of nine bars, alternately of steel and brass. These are so ar-
ranged that the steel bars, being fastened at the top, have to ex-
pand downward ; while the brass ones, fastened at the bottom,
expand upward. The expansive power of brass is to that of steel
as 100 to 61 ; therefore, if the length of the steel bars is made
10 %i the length of the brass bars, the expansion of the one metal
counterbalances that of the other, and the pendulum always re-
mains of the same length. The steel bars in the figure are rep-
resented by heavy black lines ; the brass ones, by close parallel
lines.

147. A clock is regulated by lengthening or shortening its
pendulum. This is done by screwing the ball up or down on
the rod. The ball is lowered when the clock goes too fast, and

GRIDIRON raised when it goes too slow.

PENDULUM.



EXAMPLES FOR PRACTICE.

1. (See Fig. 45, and 107, 109.) What would be the weight (that is, the

measure of the earth's attraction) of an iceberg containing 40,000 tons of
ice, if raised to a height of 1,000 miles above the earth's surface?
What would it weigh 1,000 miles beneath the earth's surface?

2. A horse at the earth's surface weighs 1,200 pounds ; what would he weigh

4,000 miles above the surface ?

How far beneath the surface would he have to be sunk, to have the
same weight ?

8. A Turkish porter will carry 800 pounds ; how many such pounds could he
carry, if he were placed half way between the surface and the centre of
the earth, and retained the same strength ? Ans. 1,600.

How many such pounds could he carry, if elevated 4,000 miles above
the surface with the same strength?

time? What will cause it to vibrate more rapidly, and what less? 145. To prevent
a clock from being affected by heat and cold, Avhat is used? Describe the Gridiron
Pendulum. 147. How is a clock regulated ?



EXAMPLES FOR PRACTICE. 69

4. "What would a body weighing -100 pounds at the earth's surface weigh

1,000 miles above the surface ?
What would it weigh 1,000 miles below the surface ?

5. "Would an IS- pound cannon-ball weigh more or less, 2,000 miles above the

earth's surface, than 2,000 miles below it, and how much ?

6. At the centre of the earth, what would be the difference of weight between

a man weighing 200 pounds at the surface and one weighing 100 pounds ?
Four thousand miles above the surface, what would be the difference
in their weight ?

7. (See Rule 1, 121. In the examples that follow ', no allowance is made for

the resistance of the air,} A man falls from a church steeple ; how many
feet will he pass through in the third second of his descent ?
6. How far will a stone fall in the twelfth second of its descent?

9. (See Rule 2, 121.) How great a velocity does a falling stone attain in 7

seconds ?

10. A hail-stone has been falling one-third of a minute; what is its velocity?

11. (See Rale 3, 121.) How far will a stone fall in 10 seconds?

12. How far will a hail-stone fall in one-third of a minute?

13. I drop a pebble into an empty well, and hear it strike the bottom in ex-
actly two seconds. How deep is the well ?

How many feet does the pebble fall in the first second of its descent?
How many, in the second ?

What velocity has the pebble at the moment of striking ?

14. A musket-ball dropped from a balloon continues falling half a minute be-
fore it reaches the earth ; how high is the balloon, and what is the velo-
city of the ball when it reaches the earth ?

15. What is the velocity of a stone dropped into a mine, after it has been fall-
ing 7 seconds, and how far has it descended ?

16. (See 122.) What would be the velocity of the same stone at the end of
the seventh second, if thrown into the mine with a velocity of 20 feet in
a second, and how far would it have descended ?

17. An arrow falls from a balloon for 9 seconds. How far does it fall alto-
gether, how far in the last second, and what velocity does it attain?

What would these three answers be, if the arrow were discharged from
the balloon with a velocity of 10 feet per second?

18. (See 125.) How long will a ball projected upwards with a Telocity of
123 2 /3 feet per second, continue to ascend ?

How great a height will it attain ?

What will be its velocity, after it has been ascending one second ?
After two seconds ? After three seconds ?

19. How many seconds will a musket-ball, shot upward with a velocity of
225V 6 feet in a second, continue to ascend?

How many feet will it rise ?

20. A stone thrown up into the air rises two seconds ; with what velocity was
it thrown?

21. (See Ml.) How many times longer must a pendulum be, to vibrate only
once in a second, than to vibrato three times in a second ?



O MECHANICS.

22. Two pendulums at the Cape of Good Hope vibrate respectively in 40 sec-
onds and 10 seconds ; how many times longer is the one than the other ?

23. Two pendulums at New Orleans vibrate in 40 seconds and 10 seconds ;
how many times longer is one than the other?

24. In the latitude of New York, a pendulum vibrating seconds is SO 1 /^
inches in length ; how long must one be, to vibrate once in 10 seconds ?
Ans. 3,910 inches.

How long must one be, to vibrate 4 times in a second at the same place ?
Ans. 2 7I / 160 inches.

25. At the equator, a pendulum 39 inches long vibrates once in a second ; how
long must a pendulum be, to vibrate once in half an hour at the same
place ?

How long must one be, to vibrate 10 times in a second ?

26. At Trinidad, a seconds-pendulum must be about 39Vso inches long ; what
would be the length of one that would vibrate 3 times in a second?

What would be the length of one that would vibrate 3 times in a
minute ?
What would be the length of one that would vibrate 3 times in an hour ?



CHAPTER VI.

MECHANICS (CONTINUED).

CENTRE OP GRAVITY.

148. THE Centre of Gravity of a body is that point
about which all its parts are balanced.

The centre of gravity is nothing more than the centre
of weight. Cat a body of uniform density in two, by a
plane passed in any direction through its centre of gravity,
and the parts thus formed will weigh exactly the same.
The whole weight of a body may be regarded as concen-
trated in its centre of gravity.

149. The Centre of Gravity must be carefully distin-
guished from the Centre of Magnitude and the Centre of
Motion.

143. What is the Centre of Gravity? How may we divide a body of uniform
density into two parts of equal weight ? Where may we regard the whole weight of
a body as concentrated ? 149. From what must the centre of gravity be carefully



CENTEE OF GRAVITY.




150. The Centre of Magnitude (or, as we briefly call it,
the Centre) of a body, is a point equally distant from its
opposite sides.

151. The Centre of Motion in a revolving surface is a
point which remains at rest, while all the other points of
the surface are in motion.

In all revolving bodies, a number of points remain at
rest. The line connecting them is called the Axis of
Motion, or briefly, the Axis of the body.

152. The centre of gravity may coincide Fig. 56.

with the centre of magnitude and lie in the
axis of motion, but need not do so. In
Fig 56, A represents a wheel entirely of
wood of uniform density ; here the centre
of gravity coincides with the centre of
magnitude, C, and both lie in the axis of
motion. B represents the same wheel
with its two lower spokes and part of the felly of lead. The centre of
magnitude, C, still lies in the axis, but the centre of gravity has fallen
to D.

When a body is of uniform density, its centre of gravity coincides with
its cent :e of magnitude. When one part of a body is heavier than another,
the centre of gravity lies nearer the heavier part.

153. A line drawn perpendicularly downward from the
centre of gravity is called the Line of Direction. In Fig.
56, CE and D E are the Lines of Direction.

154. Hov, r TO FIXD THE CENTRE OP GRAVITY. The part
of a body in which the centre of gravity is situated, may be
found, in some cases, by balanc-
ing it on a point. Thus the cen-
tre of gravity of the poker rep-
resented in Fig. 57 lies directly

over the point on which it is
balanced.

155. In a solid of regular

distinguished ? 15D. What is the Centre of Magnitude ? 151. What is the Centre of
Motion? What is the Axis of a revolving sphere? 152. Show, with Fig. 56, how the
centre of gravity may not coincide with the centre of magnitude, or lie in the axis.
When does a body's centre of gravity coincide with its centre of magnitude? When
o;i" purl is heavier than another, where does the centre of gravity lie? 153.




72



MECHANICS.



Fig. 53.



shape and uniform thickness and density, so thin that it
may be regarded as a mere surface, such as a piece of paste-
board, the centre of gravity may be found by ascertaining
any two straight lines drawn from side to side that will
divide it into two equal parts. The point at which these
lines intersect is the centre of gravity. Thus, in a parallel-
ogram, the centre of gravity is the point at which its two
diagonals intersect.

When such a surface is irregular in shape, sus-
pend it at any point, so that it may move freely,
and when it has come to rest, drop a plumb-line
from the point of suspension and mark its direc-
tion on the surface. Do the same at any other
point, and the centre of gravity will lie where the
two lines intersect.

Thus, suspend the irregular body represented
in Fig. 58 at the point A ; and, dropping the
plumb-line A B, mark its direction on the surface.
Then suspend it at C ; drop the plumb-line C D,
and mark its direction. The lines cross at E, and
there will be the centre of gravity.

156. When two bodies of equal
weight are connected by a rod, the
centre of gravity will be in the centre
of the rod. When two bodies of unequal weight are so con-
nected, the centre of gravity
will be nearer to the heavier
one. These principles are il-
lustrated in Fig. 59, in which
C represents the centre of
gravity.

157. STABILITY OF BODIES. The Base of a body is its
lowest side. When a body is supported on legs, like a





Fig. 59.
c



!s the Line of Direction ? 154. In some bodies, how may the part in which the cen-
tre of gravity lies be found? 155. How may the centre of gravity be found, in a thin
solid body of regular shape and uniform thickness and density ? How may it be
found in such a solid, whon the shape is irregular ? Explain the process with Fig. 53.
156. When two bodies of equal weight are connected by a roil, where does the cen'ro
o?gravity lie ? How does it lie, when the bodies are of unequal weight ? 157. Wha/



CENTRE OP GRAVITY.



73



Fig. 60.



Fig. 61. Fig. 62.




chair, its base is formed by lines connecting the bottoms of
its legs.

158. When the line of direction falls within the base, a
body stands ; when not, it falls.

In Fig. 60, G is the centre of grav-
ity ; since the line of direction, G P,
falls within the base, the body will
stand. Iii Fig. 61, the line of direc-
tion falls exactly at one extremity of
the base, and the body will be over-
turned by the slightest force. In Fig. 62, the line of direction falls outside of
the base, and the body will fall.

A man carrying a load on his back naturally
bends forward, to bring his line of direction with-
in the base formed by his feet. Otherwise, the
line of direction falls outside of the base, as
shown in Fig. 63 ; and the load, if heavy, may
pull him over backward.

159. Of different bodies of the
same height, that which has the broad-
est base is the hardest to overturn, because its line of di-
rection must be moved the farthest to fall outside of its

Fig. 64.



Fig. 63.





EGYPTIAN PYRAMIDS.



is the Base of a" body? When a body is supported on legs, how is its base formed?
ir>3. How must the line of direction fall, for a body to stand? Illustrate this with
Figs. 60, 61, 62. "W hat position does a man carrying a load on his back assume, and
why? 159. Of different bodies equally high, which is the hardest to overturn?



74 MECHANICS.

base. Hence a pyramid is the most stable of all figures ;
and, of different pyramids of the same height, that which
has the broadest base is the most stable. The pyramids
of Egypt have withstood the "storms of more than three
thousand years.

The stability of stone walls is increased by making them broader at the
base than at the top. Candlesticks and inkstands generally spread out at the
bottom that they may not be easily upset. For the same reason, the legs of
chairs bend outward as they approach the floor. A three-legged stool or
table has a smaller base than one that has four legs, and is therefore more
easily upset. Hence, also, the ease with which a man standing on one leg is
overturned.

160. A ball of uniform density has its centre of gravity
at its centre of magnitude. When such a ball rests on a
level surface, the line of direction falls on the point of sup-
port, and it therefore remains in any position in which it is
placed. But, as the base of a ball consists of a single point,
the point in which it touches a level surface, a slight
push throws the line of direction beyond the base, and
causes the ball to roll.

Fig. 65. 1C1. "When a ball is placed on a

sloping surface, the line of direction
falls outside of the base, and the ball
begins to roll. A cube placed on the
same sloping surface maintains its po-
sition, because the line of direction
falls within its base. See Fig. G5, in




which C represents the centre of gravity.

162. Of different bodies with bases equally large, the
lowest is the hardest to overturn, because its line of direc-
tion is least liable to fall outside of its base.



Why? What is the most stable of all figures? TIow long have the pyramids of
Egypt stood ? Give some familiar instances in which the base of a body is inado
larger than the top, to increase its stability. Why are threc-lcggi'd chairs and tables
easily overturned ? 160. In a. ball of uniform density, where is the centre of gravity ?
What is said of the stability of such aball,when resting on a level surface ? 161. When
such a ball is placed on a sloping surface, what takes place ? Compare it, in this re-
epect, with a cube. 162. Of different bodies with bases equally large, which is tho



CENTRE OF GRAVITY.



This is apparent from Figs. 66 and
67. The unfinished tower, though
leaning far over, maintains its upright



Fig. 66.





position, the
line of direc-
tion falling
within the
base. When
made higher
by the addi-
tion of seve-
ral stories, as

shown in Fig. 67, it will fall, because

the centre of gravity has been raised,

and the line of direction now falls outside of the base.

High chairs for children are unsafe, unless their legs spread at the bot-
tom. A coach with

heavy baggage piled

on its top is in danger

of upsetting on a

rough road. On the X^/ \r^

same principle, a load

of stone may pass safe-
ly over a hill-side, on

which a load of hay

would be overturned.

Fig. 68 shows that the

line of direction in the

one case would fall

within the base, while

in the other it would

fall outside of it.

163. The lower its centre of gravity, the more stable a
body is. Those, therefore, who pack goods in wagons or
vessels, should place the heaviest lowest.

This principle has been turned to account in building leaning towers. The
tower of Pisa, which is the most remarkable of these structures, with a height
of 150 feet, leans to such a degree that its top overhangs its base more than
12 feet ; yet it has stood firm for centuries. In this case, the centre of grav-
ity has been brought lower than it would otherwise have been, by the use of
heavy materials at the bottom and lighter ones higher up. The lower stories
are of dense volcanic rock, the middle stories of brick, and the upper ones of

hardest to overturn ? Why ? Illustrate this point with Figs. 66 and 67. Give some
familiar applications of this principle. 163. Why do those who pack goods in wagons
placo the heaviest lowest ? In what has this principle been turned to account ? De-




76



MECHANICS.




Fig. 70.



an exceedingly porous stone. Thus built, the tower is much less liable to
fall, than if the same material had been used throughout.

164. When the centre of gravity is brought beneath the
point of support, the stability of a body
is still further increased.

This is shown in Fig. 69. To balance a needle on
its point is next to impossible, on account of the
smallness of the base, and the height of the centre of
gravity. It may be done, however, by running the
head of the needle into a piece of cork, C, and stick-
ing into opposite sides of this cork two forks, A, B, at
equal angles. The whole may then be poised upon
the needle's point on the bottom of a wine-glass. In
this case, the heavy handles of the forks bring the
centre of gravity below the point of support, in the
stem of the glass.

The common toy known as the Rocking
Horse, represented in Fig. 70, is made on this
principle. To a horse of any light material,
bearing a trooper or some other figure, is at-
tached a wire to which a ball may be fastened.
TVhen the hind legs of the horse are placed on
the stand without the ball, the line of direction
falls outside of the base, and the horse and his
rider fall. When the ball is attached, how-
ever, the centre of gravity is brought below
the point of support ; the horse will then main-
tain its upright position, and by moving the
ball may be made to rock up and down.

165. EFFECT OF ROTARY MO-
TION. Rotary Motion, that is, mo-
tion round an axis, may keep a body from falling, even
when its line of direction falls outside of its base. Thus, if
a boy tries to balance his top on its point, he finds it im-
possible ; but, when he spins it, it stands as long as the ro-
tary motion continues. The centre of gravity is not over
the point of support all the time the top is spinning, but is




EOCKING-HOESE.



scribe the tower of Pisa, and the materials of which it is bnilt. 1G4. now is the sta-
bility of a body further increased ? Show how a needle may be balanced on its point
by applying this principle. Describe the Rocking Horse, and explain the principle
involved. 165. What is meant by Eotary Motion ? What is one of its effects ? Why
does a top fall over when we try to balance it on its point, but not fall when spinning?



CENTRE OF GRAVITY IN MAN.



constantly moving round the axis of motion ; and, beforo
the top can fall in Consequence of its being on one side of
the axis, it reaches the other side, and thus counteracts the
previous impulse. Hence, the faster the top revolves, the
steadier it is ; as its motion slackens, it gradually reels more
and more, and finally falls.

166. CENTRE OF GRAVITY IN MAN. The centre of
gravity in the body of a man lies between the hips ; the
base is formed by lines connecting the extremities of the
feet. A man enlarges this base, and therefore stands more
firm, when he turns his toes out and places his feet a short
distance apart. When old and infirm, he enlarges his base



Online LibraryG. P. (George Payn) QuackenbosA natural philosphy: embracing the most recent discoveries in the various branches of physics .. → online text (page 7 of 42)