FORCE AND MOTION
(1) FORCE, HOW MEASURED AND REPRESENTED
=71. Force.=--We have been studying various forces, such as air
pressure, pressure in liquids, and the force of elasticity in solids,
and have considered them simply as pushes or pulls. A more formal study
of forces in general and of devices for representing and measuring them
will be helpful at this point of the course.
_A force is that which tends to cause a change in the size or shape of a
body or in its state of motion._ In other words a force is a push or a
pull. That is, force tends to produce distortion or change of motion in
a body. Force itself is invisible. We measure it by the effect it
produces. Forces are usually associated with the objects exerting them.
Thus we speak of _muscular force_, _air pressure_, _liquid pressure_,
the force of a spring, the force of the earth's attraction and so on.
Forces are classified in various ways.
I. With respect to the _duration and steadiness_ of the force.
(a) Constant, as the earth's attraction. (b) Impulsive, as the stroke of
a bat on a ball. (c) Variable, as the force of the wind.
II. With respect to the _direction_ of the force.
(a) Attractive, as the earth's attraction. (b) Repulsive, as air
pressure, liquid pressure, etc.
=72. Methods of Measuring Force.=--Since forces are measured by their
effects which are either distortion or change of motion, either of these
effects may be used to measure them. For example, the force exerted by
a locomotive is sometimes computed by the _speed_ it can develop in a
train of cars in a given time, or the force of the blow of a baseball
bat is estimated by the _distance_ the ball goes before it strikes the
ground.
The more common method of measuring force, however, is by _distortion_,
that is, by measuring the change of shape of a body caused by the force.
In doing this, use is made of Hooke's Law (Art. 32), in which it is
stated that "within the limits of perfect elasticity," changes of size
or shape are directly proportional to the forces employed. That is,
twice as great a force will produce twice as great a change of shape and
so on.
[Illustration: FIG. 55.--A spring balance.]
A common contrivance using this principle is the spring balance (Fig.
55), with which all are familiar, as ice scales, meat scales, postal
scales, etc. The object which changes shape in this device is a coiled
spring contained in the case of the instrument. The balance is so
constructed that when the spring is pulled out as far as possible it has
not reached its limit of elasticity, since, if the spring were stretched
so as to exceed its elastic limit, the index would not return to its
first position on removing the load. (See Arts. 30-32.)
=73. Graphic Representation of Forces.=--A force is said to have three
elements. These are (a) _its point_ of _application_, (b) _its
direction_, and (c) _its magnitude_. For example, if there is hung upon
the hook of a spring balance a weight of 5 lbs., then we have: (a) its
point of application on the hook of the balance, (b) its downward
direction and (c) its magnitude, or 5 lbs. These three elements may be
represented by a line. Thus in Fig. 56_a_, a line _AB_ is drawn as
shown, five units long; _A_ represents the point of application; _B_,
the arrow head, shows the direction; and the length of the line (five
units) shows the magnitude of the force.
This is called a _graphic representation_ since it represents by a line
the quantity in question. If another weight of 5 lbs. were hung from the
first one, the graphic representation of both forces would be as in Fig.
56_b_. Here the first force is represented by _AB_ as before, _BC_
representing the second force applied. The whole line represents the
_resultant_ of the two forces or the result of their combination. If the
two weights were hung one at each end of a short stick _AC_ (Fig.
56_c_), and the latter suspended at its center their combined weight or
_resultant_ would of course be applied at the center. The direction
would be the same as that of the two weights. The resultant therefore is
represented by _ON_. In order to exactly balance this resultant _ON_, a
force of equal magnitude but opposite in direction must be applied at
the point of application of _ON_, or _O_. _OM_ then represents a force
that will just balance or hold in equilibrium the resultant of the two
forces _AB_ and _CD_. This line _OM_ therefore represents the
_equilibrant_ of the weights _AB_ and _CD_. The resultant of two forces
at an angle with each other is formed differently, as in Fig. 57 _a_.
Here two forces _AB_ and _AC_ act at an angle with each other. Lay off
at the designated angle the lines _AB_ and _AC_ of such length as will
accurately represent the forces. Lay off _BD_ equal to _AC_ and _CD_
equal to _AB_. The figure _ABCD_ is then a parallelogram. Its diagonal
_AD_ represents the resultant of the forces _AB_ and _AC_ acting at the
angle _BAC_. If _BAC_ equals 90 degrees or is a right angle, _AD_ may be
_computed_ thus: _AB² + BD² = AD²_. Why?
and _AD_ = √_([line]AB² + [line]BD²)._
[Illustration: FIG. 56.--Graphic representation of forces acting along
the same or parallel lines.]
[Illustration: FIG. 57.--Graphic representation of two forces acting
(_a_) at a right angle, (_b_) at an acute angle.]
This method of determining the resultant by _computation_ may be used
when the two forces are at right angles. (In any case, _AD may be
measured_ using the same scale that is laid off upon _AB_ and _AC_, as
shown in Fig. 57 _b_.) The three cases of combining forces just given
may be classified as follows: The _first_ is that of _two forces acting
along the same line_ in the same or opposite direction, as when two
horses are hitched tandem, or in a tug of war. The _second_ is that of
_two forces acting along parallel lines_, in the same direction, as when
two horses are hitched side by side or abreast. The _third_ is that of
_two forces acting at the same point at an angle_. It may be represented
by the device shown in Fig. 58, consisting of two spring balances
suspended from nails at the top of the blackboard at _A_ and _B_. A cord
is attached to both hooks and is passed through a small ring at _O_ from
which is suspended a known weight, _W_. Lines are drawn on the
blackboard under the stretched cords, from _O_ toward _OA_, _OB_, and
_OW_ and distances measured on each from _O_ to correspond to the three
forces as read on balance _A_ and _B_ and the weight _W_. Let a
parallelogram be constructed on the lines measured off on _OA_ and _OB_.
Its diagonal drawn from _O_ will be found to be vertical and of the same
length as the line measured on _OW_. The diagonal is the _resultant_ of
the two forces and _OW_ is the equilibrant which is equal and opposite
to the resultant.
[Illustration: FIG. 58.--Experimental proof of parallelogram of forces.]
Again, the _first_ case may be represented by a boat moving up or down a
stream; the resultant motion being the combined effect of the boat's
motion and that of the stream. The _second_, may be represented by two
horses attached side by side to the same evener. The resultant force
equals the sum of the two component forces. The _third_, may be
represented by a boat going across a stream, the resultant motion being
represented by the diagonal of the parallelogram formed by using the
lines that represent the motion of the stream and of the boat.
=74. Units for Measuring Force.=--Force is commonly measured in units of
weight: in pounds, kilograms, and grams. For example, we speak of 15
lbs. pressure per square inch and 1033.6 g. pressure per square
centimeter as representing the air pressure. It should be noted here
that the words pound, kilogram, and gram are used not only to represent
_weight_ or _force_ but also the masses of the objects considered. Thus,
one may speak of a pound-mass meaning the amount of material in the
object.
It will help to avoid confusion if we reserve the simple terms "gram"
and "pound" to denote exclusively an amount of matter, that is, a mass,
and to use the full expression "gram of force" or "pound of force"
whenever we have in mind the pull of the earth upon these masses. Or,
one may speak of a _pound-weight_ meaning the amount of attraction
exerted by the earth upon the object. The same is true of _gram-mass_
and _gram-weight_. The mass of a body does not change when the body is
transferred to another place. The weight, however, may vary, for on
moving a body from the equator toward the poles of the earth the weight
is known to increase.
Important Topics
1. Definition of force.
2. Classification of forces. (a) Duration: constant, impulsive,
variable. (b) Direction: attractive, repulsive.
3. Methods of measuring force. (a) By distortion. (b) By change of
motion.
4. Graphic representation of forces: component, resultant, equilibrant.
5. Three cases of combining forces. (1) Two forces acting on the same
line. (2) Two forces acting in parallel lines. (3) Two forces acting at
the same point at an angle.
6. Units for measuring force, pound, gram.
Exercises
1. Name five natural forces. Which produce a tension? Which a pressure?
2. How much can you lift? Express in pounds and kilograms.
3. Show graphically the resultant of two forces at right angles, one of
12 lbs., the other of 16 lbs. What is the magnitude of this resultant?
Then determine the answer, first by measurement and then by computation.
Which answer is more accurate? Why?
4. Represent by a parallelogram the two forces that support a person
sitting in a hammock and draw the line representing the resultant.
5. Find graphically the resultant of the pull of two forces, one of 500
lbs. east and one of 600 lbs. northwest.
6. Determine the equilibrant of two forces, one of 800 lbs. south and
one of 600 lbs. west.
7. Would the fact that weight varies on going from the equator to either
pole be shown by a spring balance or a beam balance? Explain.
(2) MOTION. NEWTON'S LAWS OF MOTION
=75. Motion a Change of Position.=--Motion is defined as a continuous
_change in the position_ of a body. The _position_ of a body is usually
described as its _distance_ and _direction_ from some fixed point. Thus
a man on a boat may be at rest with respect to the boat and moving with
respect to the earth. Or, if he walks toward the stern as fast as the
boat moves forward, he may keep directly over a rock on the bottom of
the lake and hence not be moving with reference to the rock and yet be
in motion with respect to the boat. Motion and rest, therefore, are
_relative_ terms. The earth itself is in motion in turning on its axis,
in moving along its orbit, and in following the sun in its motion
through space. Motions are classified in several ways:
(A) MODES OF MOTION
1. _Translation._--A body is said to have motion of _translation_ when
every line in it keeps the same direction.
2. _Rotation._--A body has motion of _rotation_ when it turns upon a
fixed axis within the body, as a wheel upon its axle or the earth upon
its axis.
3. _Vibration_ or _Oscillation_.--A body is said to have _vibratory_ or
_oscillatory_ motion when it returns to the same point at regular
intervals by reversals of motion along a given path, _e.g._, a pendulum.
(B) DIRECTION OF MOTION
1. _Rectilinear._--A body has rectilinear motion when its path is a
straight line. Absolute rectilinear motion does not exist, although the
motion of a train on a straight stretch of track is nearly rectilinear.
2. _Curvilinear._--A body has _curvilinear_ motion when its path is a
curved line, _e.g._, the path of a thrown ball.
(C) UNIFORMITY OF MOTION
1. _Uniform._--A body has uniform motion when its speed and direction of
motion do not change. Uniform motion for extended periods is rarely
observed. A train may cover, on an average, 40 miles per hour but during
each hour its speed may rise and fall.
2. _Variable._--A body has variable motion when its speed or direction
of motion is continually changing. Most bodies have variable motion.
3. _Accelerated._--A body has accelerated motion when its speed or
direction of motion continually changes. If the speed changes by the
same amount each second, _and the direction of motion does not change_
the motion is said to be _uniformly_ accelerated, _e.g._, a falling
body.
Uniformly accelerated motion will be studied further under the topic of
falling bodies.
_Velocity_ is _the rate of motion_ of a body in a given direction. For
example, a bullet may have a velocity of 1300 ft. a second upwards.
_Acceleration_ is _the rate of change of velocity_ in a given direction,
or the change of velocity in a unit of time. A train starting from a
station gradually increases its speed. The gain in velocity during one
second is its acceleration. When the velocity is decreasing, as when a
train is slowing down, the acceleration is opposite in direction to the
velocity. A falling body falls faster and faster. It has _downward
acceleration_. A ball thrown upward goes more and more slowly. It also
has _downward acceleration_.
=76. Momentum.=--It is a matter of common observation that a heavy body
is set in motion with more difficulty than a light one, or if the same
force is used for the same length of time upon a light and a heavy
body,[E] the light body will be given a greater velocity. This
observation has led to the _calculation_ of what is called the "quantity
of motion" of a body, or its _momentum_. It is computed by multiplying
the mass by the velocity. If the C.G.S. system is used we shall have as
the momentum of a 12 g. body moving 25 cm. a second a momentum of 12 ×
25 or 300 C.G.S. units of momentum. This unit has no name and is
therefore expressed as indicated above. The formula for computing
momentum is: _M = mv_.
[E] By a light body is meant one of small mass, a heavy body
possessing much greater mass.
Newton's Laws of Motion
=77. Inertia, First Law of Motion.=--One often observes when riding in a
train that if the train moves forward suddenly the passengers do not get
into motion as soon as the train, and apparently are jerked backward.
While if the train is stopped suddenly, the passengers tend to keep in
motion. This tendency of matter to keep moving when in motion and to
remain at rest when at rest is often referred to as the property of
_inertia_. _Newton's first law of motion_, often called the _law of
inertia_, describes this property of matter as follows:
_Every body continues in a state of rest or of uniform motion in a
straight line unless it is compelled to change that state by some
external force._ This means that if an object like a book is lying on a
table it will remain there until removed by some outside force. No
inanimate object can move itself or stop itself. If a ball is thrown
into the air it would move on forever if it were not for the _force_ of
attraction of the earth and the resistance of the air.
It takes time to put a mass into motion, a heavy object requiring more
time for a change than a light object. As an example of this, note the
movements of passengers in a street car when it starts or stops
suddenly. Another illustration of the law of inertia is the so-called
"penny and card" experiment. Balance a card on the end of a finger.
Place on it a coin directly over the finger, snap the card quickly so as
to drive the card from beneath the coin. The coin will remain on the
finger. (See Fig. 59.)
[Illustration: FIG. 59.--The ball remains when the card is driven away.]
According to Newton's first law of motion a moving body which could be
entirely freed from the action of all external forces would have uniform
motion, and would describe a perfectly straight course. The curved path
taken by a baseball when thrown shows that it is acted upon by an
outside force. This force, the attraction of the earth, is called
_gravity_.
[Illustration: SIR ISAAC NEWTON "By Permission of the Berlin
Photographic Co., New York."
Sir Isaac Newton (1642-1727) Professor of mathematics at Cambridge
university; discovered gravitation; invented calculus; announced the
laws of motion; wrote the Principia; made many discoveries in light.]
[Illustration: GALILEO GALILEI "By Permission of the Berlin Photographic
Co., New York."
Galileo Galilei (1564-1642). Italian. "Founder of experimental science";
"Originator of modern physics"; made the first thermometer; discovered
the laws of falling bodies and the laws of the pendulum; invented
Galilean telescope.]
[Illustration: FIG. 60.--Cross-section of the DeLaval cream separator.]
=78. Curvilinear Motion.=--Curvilinear motion occurs when a moving body
is pulled or pushed away from a straight path. The pull or push is
called _centripetal_ (center-seeking) force. A moving stone on the end
of a string when pulled toward the hand moves in a curve. If the string
is released the stone moves in a tangent to the curve. The string pulls
the hand. This phase of the pull is called _centrifugal_ force. The
_centripetal_ force is the pull on the stone. Centripetal and
centrifugal force together cause a tension in the string. Examples of
curvilinear motion are very common. The rider and horse in a circus ring
lean inward in order to move in a curve. The curve on a running track in
a gymnasium is "banked" for the same reason. Mud flying from the wheel
of a carriage, the skidding of an automobile when passing rapidly around
a corner, and sparks flying from an emery wheel, are illustrations of
the First Law of Motion.
Cream is separated from milk by placing the whole milk in a rapidly
revolving bowl, the cream being lighter collects in the center and is
thrown off at the top. (See Fig. 60.) Clothes in steam laundries are
dried by a centrifugal drier. In amusement parks many devices use this
principle. (See centrifugal pumps, Art. 70.)
[Illustration: FIG. 61.--The two balls reach the floor at the same
time.]
=79. The Second Law of Motion,= sometimes called the _law of momentum_,
leads to the _measurement of force_, by the momentum or the quantity of
motion, produced by it. The law is stated as follows:
_Change of motion, or momentum, is proportional to the acting force and
takes place in the direction in which the force acts._ In other words,
if two or more forces act at the same instant upon a body each produces
the same effect that it would if acting alone. If a card be supported on
two nails driven horizontally close together into an upright board (see
Fig. 61), and two marbles be so placed on the ends as to balance each
other, when one marble is snapped horizontally by a blow, the other will
fall. Both reach the floor at the same time. The two balls are equally
pulled down by the earth's attraction and strike the ground at the same
time, though one is shot sidewise, and the other is dropped vertically.
As gravity is a constant force, while the blow was only a momentary
force, the actual path or resultant motion will be a curved line.
The constant relation, between the acting force and the change of
momentum it produces in a body, has led to the adoption of a convenient
C.G.S. unit of force called the _dyne_. _The dyne is that force which
can impart to a mass of one gram a change of velocity at the rate of one
centimeter per second every second._ This definition assumes that the
body acted upon is free to move without hindrance of any kind, so that
the acting force has to overcome only the _inertia_ of the body.
_However_, the _law_ applies in every case of application of force, so
that each force produces its full effect independently of other forces
that may be acting at the same time upon the body.
=80. Newton's Third Law.=--This law has been experienced by everyone who
has jumped from a rowboat near the shore. The muscular action that
pushes the body forward from the boat also pushes the boat backward,
often with awkward results. The law is stated: _To every action, there
is always an opposite and equal reaction, or the mutual actions of any
two bodies are always equal and opposite in direction_. Many
illustrations of this law are in every one's mind: a stretched rope
pulls with the same force in one direction as it does in the opposite
direction. If a bat hits a ball, the ball hits the bat with an equal and
opposite force. The third law is therefore sometimes called the law of
_reaction_. When a weight is hung upon a spring balance the action of
the weight pulls down the spring until it has stretched sufficiently
(Hooke's Law) to produce an elastic _reaction_ that equals and hence
supports the weight. When a man stands at the center of a plank
supported at its ends, the action of the man's weight bends the plank
until the elastic force developed in the plank equals the weight
applied. Further, when a train or a wagon is on a bridge the bridge
yields until it has developed an elastic reaction equal to the weight
applied. If a person stands in the center of a room, the floor beams
yield until the third law is satisfied. In fact, whenever a force acts,
a contrary equal force always acts.
=81. Stress and Strain.=--A pair of forces that constitute an action and
a reaction is called a _stress_. The two forces are two parts of one
_stress_. If the two forces act away from each other, as in the breaking
of a string, the stress is called a _tension_, but if they act toward
each other as in crushing anything, the stress is called a _pressure_.
In order for a body to exert force it must meet with resistance. The
force exerted is never greater than the resistance encountered. Thus one
can exert but little force upon a feather floating in the air or upon
other light objects. A fast moving shot exerts no force unless it
encounters some resistance.
Forces, then, are always found in pairs. Thus to break a string, to
stretch an elastic band, to squeeze a lemon, one must exert two equal
and opposite forces. Such a thing as a single force acting alone is
unknown. Usually, however, we give our attention mainly to one of the
forces and ignore the other. When a force acts upon a body the change of
shape or size resulting is called a _strain_. Hooke's law (Art. 32) is
often expressed as follows: "The strain is proportional to the stress,"
_e.g._, the stretch of the spring of a spring balance is proportional to
the load placed upon it.
Important Topics
1. Motion a change of position. Kinds of motion.
2. Newton's laws of motion.
3. Momentum.
4. Inertia. First law of motion. Curvilinear motion.
5. Second law of motion.
6. Third law of motion. Action and reaction, stress and strain.
Exercises
1. Mention three illustrations of the third law, different from those
given.
2. A rifle bullet thrown against a board standing upon edge will knock
it down; the same bullet fired at the board will pass through it without
disturbing its position. Explain.
3. A hammer is often driven on to its handle by striking the end of the
latter. Explain.
4. Consider a train moving 60 miles an hour, with a gun on the rear
platform pointing straight backward. If a ball is fired from the gun
with a speed of 60 miles an hour, what will happen to the ball?
5. Could one play ball on the deck of an ocean steamer going 25 miles an
hour without making allowance for the motion of the ship? Explain.
6. On a railroad curve, one rail is always higher. Which? Why?
7. Why can a small boy when chased by a big boy often escape by dodging?
8. Will a stone dropped from a moving train fall in a straight line?
Explain.
9. A blast of fine sand driven against a sheet of glass soon gives it a
rough surface. Explain.
10. Explain the use of fly-wheels in steadying the motion of machinery
(for example, the sewing machine).
11. Is it easier to walk to the front or rear of a passenger train when
it is stopping? Why?
12. Why does lowering the handles of a wheel-barrow on the instant of
striking make it easier to go over a bump?
13. Why should a strong side wind interfere with a game of tennis? How
can it be allowed for?
14. On which side of a railroad track at a curve is it the safer to walk
while a train is passing? Why?
15. Why does a bullet when fired through a window make a clean round
hole in the glass, while a small stone thrown against the window
shatters the glass?
16. A tallow candle can be fired through a pine board. Why?
17. In cyclones, straws are frequently found driven a little distance
into trees; why are the straws not broken and crushed instead of being
driven into the tree unbroken?
18. A bullet weighing one-half oz. is fired from a gun weighing 8 lb.
The bullet has a velocity of 1800 ft. per second. Find the velocity of
the "kick" or recoil of the gun.
18. When football players run into each other which one is thrown the
harder? Why?
20. A railroad train weighing 400 tons has a velocity of 60 miles per
hour. An ocean steamer weighing 20,000 tons has a velocity of one half
mile per hour. How do their momenta compare?
21. Why is a heavy boy preferable to a lighter weight boy for a football
team?
22. Why does a blacksmith when he desires to strike a heavy blow, select
a heavy sledge hammer and swing it over his head?
23. Why does the catcher on a baseball team wear a padded glove?
(3) RESOLUTION OF FORCES
=82. Resolution of Forces.=--We have been studying the effect of forces
in producing motion and the results of combining forces in _many_ ways;
in the _same line_, in _parallel lines_, and in _diverging lines_.
Another case of much interest and importance is _the determination of
the effectiveness of a force in a direction different from the one in
which it acts_. This case which is called _resolution of forces_ is
frequently used. To illustrate: one needs but to recall that a sailor
uses this principle in a practical way whenever he sails his boat in any
other direction than the one in which the wind is blowing, _e.g._, when
the wind is blowing, say from the north, the boat may be driven east,
west, or to any point south between the east and west and it is even
possible to beat back against the wind toward the northeast or
northwest. Take a sled drawn by a short rope with the force applied
along the line _AB_ (see Fig. 62); part of this force tends to lift the
front of the sled as _AC_ and a part to draw it forward as _AD_. Hence
not all of the force applied along _AB_ is used in drawing the sled
forward. Its effectiveness is indicated by the relative size of the
component _AD_ compared to _AB_.
[Illustration: FIG. 62.--_AD_ is the effective component.]
The force of gravity acting upon a sphere that is resting on an
_inclined plane_ may be readily resolved into two components, one, the
_effective_ component, as _OR_, and the other, the _non-effective_ as
_OS_. (See Fig. 63.) If the angle _ACB_ is 30 degrees, _AB_ equals 1/2
of _AC_ and _OR_ equals 1/2 of _OG_, so that the speed of the sphere
down the plane developed in 1 second is less than (about one-half of)
the speed of a freely falling body developed in the same time. Why is
_OS_ non-effective?
[Illustration: FIG. 63.--The effective component is _OR_.]
[Illustration: FIG. 64.--Resolution of the forces acting on an
aeroplane.]
=83. The Aeroplane.=--The aeroplane consists of one or two frames _ABCD_
(see Fig. 64), over which is stretched cloth or thin sheet metal. It is
driven through the air by a propeller turned by a powerful gasoline
motor. This has the effect of creating a strong breeze coming toward the
front of the aeroplane. As in the case of the sailboat a pressure is
created at right angles to the plane along _GF_ and this may be resolved
into two components as _GC_ and _GE_, _GC_ acting to lift the aeroplane
vertically and _GE_ opposing the action of the propeller. Fig. 65
represents the Curtis Flying Boat passing over the Detroit river.
[Illustration: FIG. 65.--The Curtis hydroplane.]
Exercises.
1. If a wagon weighing 4000 lbs. is upon a hill which rises 1 ft. in 6,
what force parallel to the hill will just support the load? (Find the
effective component of the weight down the hill.)
2. If a barrel is being rolled up a 16-ft. ladder into a wagon box 3 ft.
from the ground, what force will hold the barrel in place on the ladder,
if the barrel weighs 240 lbs. Show by diagram.
3. Show graphically the components into which a man's push upon the
handle of a lawn mower is resolved.
4. Does a man shooting a flying duck aim at the bird? Explain.
5. What are the three forces that act on a kite when it is "standing" in
the air?
6. What relation does the resultant of any two of the forces in problem
five have to the third?
7. Into what two forces is the weight of a wagon descending a hill
resolved? Explain by use of a diagram.
8. A wind strikes the sail of a boat at an angle of 60 degrees to the
perpendicular with a pressure of 3 lbs. per square foot. What is the
effective pressure, perpendicular to the sail? What would be the
effective pressure when it strikes at 30 degrees?
9. How is the vertical component of the force acting on an aeroplane
affected when the front edge of the plane is elevated? Show by diagram.
(4) MOMENT OF FORCE AND PARALLEL FORCES
=84. Moment of Force.=--In the study of motion we found that the
quantity of motion is called _momentum_ and is measured by the product
of the _mass times the velocity_. In the study of _parallel forces_,
especially such as tend to produce _rotation_, we consider a similar
quantity. It is called a _moment of force_, which is the term applied to
the _effectiveness_ of a force in producing change of rotation. It also
measured by the product of two quantities; _One, the magnitude of the
force itself_, and the other, _the perpendicular distance from the axis
about which the rotation takes place to the line representing the
direction of the force_.
[Illustration: FIG. 66.--The moments about _S_ are equal.]
_To illustrate:_ Take a rod, as a meter stick, drill a hole at _S_ and
place through it a screw fastened at the top of the blackboard. Attach
by cords two spring balances and draw to the right and left, _A_ and _B_
as in Fig. 66. Draw out the balance _B_ about half way, hold it
steadily, or fasten the cord at the side of the blackboard, and read
both balances. Note also the distance _AS_ and _BS_. Since the rod is at
rest, the tendency to rotate to the right and left must be equal. That
is, the moments of the forces at _A_ and _B_ about _S_ are equal. Since
these are computed by the product of the _force times the force_ arm,
multiply _B_ by _BS_ and _A_ by _AS_ and see if the computed moments are
equal. _Hence a force that tends to turn or rotate a body to the right
can be balanced by another of equal moment that acts toward the left._
[Illustration: FIG. 67.--Law of parallel forces illustrated.]
=85. Parallel Forces.=--Objects are frequently supported by two or more
upward forces acting at different points and forming in this way a
system of parallel forces; as when two boys carry a string of fish on a
rod between them or when a bridge is supported at its ends. The
principle of moments just described aids in determining the magnitude of
such forces and of their resultant. To illustrate this take a wooden
board 4 in. wide and 4 ft. long of uniform dimensions. (See Fig. 67.)
Place several screw hooks on one edge with one set at _O_ where the
board will hang horizontally when the board is suspended there. Weigh
the board by a spring balance hung at _O_. This will be the resultant in
the following tests. Now hang the board from two spring balances at _M_
and _N_ and read both _balances_. Call readings _f_ and _f´_. To test
the forces consider _M_ as a fixed point (see Fig. 67) and the weight of
the board to act at _O_. Then the moment of the weight of the board
should be equal the moment of the force at _N_ since the board does not
move, or _w_ times _OM_ equals _f´_ times _NM_. If _N_ is considered the
fixed point then the moment of the weight of the board and of _f with
reference to the point N_ should be equal, or _w_ times _ON_ = _f_ times
_NM_. Keeping this illustration in mind, the law of parallel forces may
be stated at follows: 1. _The resultant of two parallel forces acting in
the same direction at different points in a body is equal to their sum
and has the same direction as the components._
_The moment of one of the components about the point of application of
the other is equal and opposite to the moment of the supported weight
about the other._
=Problem.=--If two boys carry a string of fish weighing 40 lbs. on
a rod 8 ft. long between them, what force must each boy exert if
the string is 5 ft. from the rear boy?
=Solution.=--The moment of the force _F_ exerted about the opposite
end by the rear boy is _F_ × 8. The moment of the weight about the
same point is 40 × (8 - 5) = 120. Therefore _F_ × 8 = 120, or _F_ =
15, the force exerted by the rear boy. The front boy exerts a force
of _F_ whose moment about the other end of the rod is _F_ × 8. The
moment of the weight about the same point is 40 × 5 = 200. Since
the moment of _F_ equals this, 200 = _F_ × 8, or _F_ = 25. Hence
the front boy exerts 25 lbs. and the rear boy 15 lbs.
[Illustration: FIG. 68.--A couple.]
=86. The Couple.=--If two equal parallel forces act upon a body along
different lines in opposite directions, as in Fig. 68, they have no
single resultant or there is no one force that will have the same effect
as the two components acting together. A combination of forces of this
kind is called a _couple_. Its tendency is to produce change of rotation
in a body. An example is the action upon a compass needle which is
rotated by a force which urges one end toward the north and by an equal
force which urges the other end toward the south.
Important Topics
1. Moment of force, how measured.
2. Parallel forces.
3. The two laws of parallel forces.
4. The couple.
Exercises
1. Show by diagram how to arrange a three-horse evener so that each
horse must take one-third of the load.
2. Two boys support a 10-ft. pole on their shoulders with a 40-lb.
string of fish supported from it 4 ft. from the front boy. What load
does each boy carry? Work by principle of moments.
3. If two horses draw a load exerting a combined pull of 300 lbs., what
force must each exert if one is 28 in. and the other is 32 in. from the
point of attachment of the evener to the load?
[Illustration: FIG. 69.--Forces acting upon a stretched rope.]
[Illustration: FIG. 70.--A crane with horizontal tie.]
4. A weight of 100 lbs. is suspended at the middle of a rope _ACB_ 20
ft. long. (See Fig. 69.) The ends of the rope are fastened at points _A_
and _B_ at the same height. Consider _D_ as the center of the line _AB_.
What is the tension of the rope when _CD_ is 3 ft.? When _CD_ is 1 ft.?
When _CD_ is 1 in.?
5. A crane is set up with the tie horizontal. (See Fig. 70.) If 1000
lbs. is to be lifted, find the tie stress and the boom stress if the
boom angle is 30 degrees? If 45 degrees? 60 degrees?
6. A ball is placed on a plane inclined at an angle of 30 degrees to the
horizontal. What fraction of its weight tends to cause motion down the
plane? What effect does the other component of the weight have? Why?
7. A person weighing 150 lbs. is lying in a hammock. The distance
between the supports is 15 ft. The hammock sags 4 ft. What is the
tension in the supports at each end? What is the tension when the sag is
only 1 ft.?
8. A ladder 30 ft. long and weighing 80 lbs. leans against the side of a
building so that it makes an angle of 30 degrees with the building. Find
the direction and magnitude of the component forces on the ground and at
the building.
9. A traveling crane 50 ft. long weighing 10 tons moves from one end of
a shop to the other, at the same time a load of 4000 lbs. moves from end
to end of the crane. Find the pressure of the trucks of the crane on the
track when the load is at a distance of 5, 10, 15, and 25 ft. from
either end.
[Illustration: FIG. 71--A truss.]
10. Resolve a force of 500 lbs. into two components at right angles to
each other, one of which shall be four times the other.
11. A truss (see Fig. 71), carries a load of 1000 lbs. at _C_. Find the
forces acting along _AC_, _BC_, and _AB_. If _AC_ and _BC_ are each 12
ft. and _AB_ 20 ft., which of these forces are tensions and which are
pressures?
(5) GRAVITATION AND GRAVITY
=87. Gravitation.=--Gravitation is the force of attraction that exists
between all bodies of matter at all distances. This attraction exists
not only between the heavenly bodies, the stars and planets, etc., but
is also found between bodies on the earth. A book attracts all objects
in a room and outside of a room as well, since its weight shows that it
is attracted by the earth itself. The gravitational attraction between
ordinary bodies is so slight that it requires careful experiments to
detect it. In fact, it is only when one of the attracting bodies is
large, as for example the earth, that the force becomes considerable.
Careful studies of the motions of the heavenly bodies, especially of
that of the moon in its orbit about the earth, led Sir Isaac Newton to
the statement of the _law of gravitation_ which is well expressed in the
following statement:
=88. Law of Gravitation.=--_Every particle of matter in the universe
attracts every other particle with a force that is directly proportional
to the product of their masses and inversely proportional to the square
of the distance between them._
The law may be separated into two parts, one referring to the masses of
the bodies concerned, the other to the effect of the distance between
them. The first part is easily understood since we all know that two
quarts of milk will weigh just twice as much as one quart. To illustrate
the second part of the law, suppose that the moon were removed to
_twice_ its present distance from the earth, then the attraction between
the earth and the moon would be _one-fourth_ its present attraction. If
removed to _three_ times its present distance, the attraction would be
_one-ninth_, etc.
The attraction of the earth for other bodies on or near it is called
_gravity_. The _weight_ of a body is the measure of the earth's
attraction for it; or it is the force of gravity acting upon it.
Newton's third law of motion states that every action is accompanied by
an equal and opposite reaction (Art. 80). Hence, the attraction of the
earth for a book or any other object is accompanied by an equal
attraction of the book for the earth.
=89. Weight.=--In advanced physics it is proved that a sphere attracts
as if it were concentrated at its center. Thus if the earth's radius be
considered as 4000 miles, then a body 4000 miles above the earth's
surface would be 8000 miles above the earth's center, or twice as far
from the center of the earth as is a body upon the earth's surface. A
body then 4000 miles above the earth's surface will weigh then but
one-fourth as much as it will at the surface of the earth.
Since the earth is flattened at the poles, the surface at the equator
is farther from the center of the earth than at points north or south.
Thus a body weighing 1 lb. at the equator weighs 1.002 lb. at Chicago,
or about 1/500 more. The rotation of the earth also affects the weight
of a body upon it so that at the equator the weight of a body is 1/289
less than at the pole. Both effects, that of flattening and of rotation,
tend to diminish the weight of bodies at the equator, so that a body at
the latter place weighs about 1/192 less than at the poles.
In studying the effect of the earth's gravity, the following
illustration will be helpful: Imagine an open shaft a mile square
extending through the earth. What would happen to a stone thrown
into the shaft? At first it would have the attraction of the whole
earth drawing it and continually increasing its speed downward. As
it descends from the surface, the pull toward the center grows less
and less. Halfway to the center the body has lost half its weight.
When the stone reaches the center, the pull in all directions is
the same, or in other words, _it has no weight_. It would, however,
continue moving rapidly on account of its inertia, and as it
continues on from the center, the greater part of the earth being
left behind, the attraction pulling toward the center will
gradually stop it. It will then fall again toward the center and be
stopped again after passing it, and after repeatedly moving up and
down will finally come to rest at the center of the earth. At this
point it will be found to be a body without weight since it is
pulled equally in all directions by the material of the earth. What
force brings the body to rest?
=90. Center of Gravity.=--A body is composed of a great many particles
each of which is pulled toward the center of the earth by the force of
gravity. A single force that would exactly equal the combined effect of
the pull of the earth for all the particles of a body would be their
resultant. The _magnitude_ of this resultant is the weight of the body.
The _direction_ of this resultant is in a line passing toward the
earth's center, while the _point of application_ of this resultant is
called the _center of gravity_ of the body. The center of gravity of a
body may also be briefly defined as _the point about which it may be
balanced_. As the location of this point depends upon the distribution
of matter in the body, the center of gravity is also sometimes called
the _center of mass_ of the body.
The earth's attraction for a body is considered for the sake of
simplicity, not as a multitude of little forces, but as a single force
applied at its center of gravity. To find the center of gravity of a
body find two intersecting lines along which it balances, see Fig. 72,
and the center of gravity will be at the intersection. A vertical line
through this point is sometimes called the _line of direction of the
weight_.
[Illustration: FIG. 72.--The center of gravity is at the intersection of
the lines of direction.]
=91. Equilibrium of Bodies.=--Equilibrium means equally balanced. A body
at rest or in uniform motion is then in equilibrium. An object is in
equilibrium under gravity when a vertical line through its center of
gravity passes through the point of support. A trunk is an example of a
body in equilibrium since a vertical line from its center of gravity
falls within the base formed by the area upon which it rests. Work will
be necessary to tip the trunk from its position. The amount of work
required will depend upon the weight of the body and the location of the
center of gravity.
=92. Kinds of Equilibrium.--(a) Stable.=--A body is in stable
equilibrium under gravity if its center of gravity is raised whenever
the body is displaced. It will return to its first position if allowed
to fall after being slightly displaced. In Fig. 73, _a_ and _b_ if
slightly tipped will return to their first position. They are in stable
equilibrium. Other examples are a rocking chair, and the combination
shown in Fig. 74.
[Illustration: FIG. 73.--Stable equilibrium.]
=(b) Unstable.=--A body is in unstable equilibrium under gravity if its
center of gravity is lowered whenever the body is slightly displaced. It
will fall farther from its first position. A pencil balanced on its
point or a broom balanced on the end of the handle are in unstable
equilibrium. The slightest disturbance will make the line of direction
of the weight fall outside of (away from) the point of support (Fig. 75
_a_).
[Illustration: FIG. 74--An example of stable equilibrium. Why?]
[Illustration: FIG. 75.--Unstable equilibrium _a_, neutral equilibrium
_b_.]
=(c) Neutral.=--A body is in neutral equilibrium if its center of
gravity is neither raised nor lowered whenever the body is moved.
Familiar examples are a ball lying on a table (Fig. 75 _b_) and a wagon
moving on a level street (referring to its forward motion).
[Illustration: FIG. 76.--_B_ is more stable than _A_.]
=93. Stability.=--When a body is in stable equilibrium, effort must be
exerted to overturn it, and the degree of stability is measured by the
effort required to overturn it. To overturn a body, it must be moved so
that the vertical line through its center of gravity will pass outside
of its supporting base. This movement in stable bodies necessitates a
raising of the center of gravity. The higher this center of gravity must
be raised in overturning the body, the more stable it is, _e.g._, see
Fig. 76. Thus a wagon on a hillside will not overturn until its weight
falls outside of its base, as in Fig. 77 _B_. The stability of a body
depends upon the position of its center of gravity and the area of its
base. _The lower the center of gravity and the larger the base_, the
more stable the body. What means are employed to give stability to
bodies, in every-day use (such as clocks, ink-stands, pitchers, vases,
chairs, lamps, etc.)?
[Illustration: FIG. 77.--_B_ will overturn; _A_ will not.]
Important Topics
1. Gravitation; law of gravitation, gravity, weight.
2. Center of gravity.
3. The three states of equilibrium. Stability.
Exercises
1. Why is a plumb-line useful in building houses?
2. What is the center of gravity of a body?
3. Explain the action of a rocking chair that has been tipped forward.
4. Is the stability of a box greater when empty or when filled with
sand? Explain.
5. How can you start yourself swinging, in a swing, without touching the
ground?
6. Is the center of gravity of the beam of a balance above, below, or at
the point of a support? How did you find it out?
7. Why are some ink bottles cone shaped with thick bottoms?
8. Would an electric fan in motion on the rear of a light boat move it?
Would it move the boat if revolving under water? Explain.
9. What turns a rotary lawn sprinkler?
10. Why, when you are standing erect against a wall and a coin is placed
between your feet, can you not stoop and pick it up unless you shift
your feet or fall over?
11. What would become of a ball dropped into a large hole bored through
the center of the earth?
12. When an apple falls to the ground, does the earth rise to meet it?
13. How far from the earth does the force of gravity extend?
14. Why in walking up a flight of stairs does the body bend forward?
15. In walking down a steep hill why do people frequently bend backward?
16. Why is it so difficult for a child to learn to walk, while a kitten
or a puppy has no such difficulty?
17. Explain why the use of a cane by old people makes it easier for them
to walk?
(6) FALLING BODIES
=94. Falling Bodies.=--One of the earliest physical facts learned by a
child is that a body unsupported falls toward the earth. When a child
lets go of a toy, he soon learns to look for it on the floor. It is also
of common observation that light objects, as feathers and paper, fall
much slower than a stone. The information, therefore, that all bodies
actually fall at the same rate in a vacuum or when removed from the
retarding influence of the air is received with surprise.
This fact may be shown by using what is called a coin and feather tube.
On exhausting the air from this tube, the feather and coin within are
seen to fall at the same rate. (See Fig. 78.) when air is again
admitted, the feather flutters along behind.
[Illustration: FIG. 78.--Bodies fall alike in a vacuum.]
=95. Galileo's Experiment.=--The fact that bodies of different weight
tend to fall at the same rate was first experimentally shown by Galileo
by dropping a 1-lb. and a 100-lb. ball from the top of the leaning tower
of Pisa in Italy (represented in Fig. 79). Both starting at the same
time struck the ground together. Galileo inferred from this that
feathers and other light objects would fall at the same rate as iron or
lead were it not for the resistance of the air. After the invention of
the air pump this supposition was verified as just explained.
[Illustration: FIG. 79.--Leaning tower of Pisa.]
=96. Acceleration Due to Gravity.=--If a body falls freely, that is
without meeting a resistance or a retarding influence, its motion will
continually increase. The _increase_ in motion is found to be constant
or uniform during each second. This uniform increase in motion or in
velocity of a falling body gives one of the best illustrations that we
have of uniformly accelerated motion. (Art. 75.) On the other hand, a
body thrown upward has uniformly retarded motion, that is, its
acceleration is downward. The velocity acquired by a falling body in
unit time is called its _acceleration_, or the _acceleration due to
gravity_, and is equal to 32.16 ft. (980 cm.) per second, downward, each
second of time. In one second, therefore, a falling body gains a
velocity of 32.16 ft. (980 cm.) per second, downward. In two seconds it
gains twice this, and so on.
In formulas, the acceleration of gravity is represented by "_g_" and the
number of seconds by _t_, therefore the formula for finding the
velocity, _V_,[F] of a falling body starting from rest is _V_ = _gt_. In
studying gravity (Art. 89) we learned that its force varies as one moves
toward or away from the equator. (How?) In latitude 38° the acceleration
of gravity is 980 cm. per second each second of time.
[F] _V_ represents the velocity of a falling body at the end of _t_
seconds.
=97. Experimental Study of Falling Bodies.=--To study falling bodies
experimentally by observing the fall of unobstructed bodies is a
difficult matter. Many devices have been used to reduce the motion so
that the action of a falling body may be observed within the limits of a
laboratory or lecture room. The simplest of these, and in some respects
the most satisfactory, was used by Galileo. It consists of an inclined
plane which reduces the effective component of the force of gravity so
that the motion of a body rolling down the plane may be observed for
several seconds. For illustrating this principle a steel piano wire has
been selected as being the simplest and the most easily understood. This
wire is stretched taut across a room by a turn-buckle so that its slope
is about one in sixteen. (See Fig. 80.) Down this wire a weighted pulley
is allowed to run and the distance it travels in 1, 2, 3, and 4 seconds
is observed. From these observations we can compute the distance covered
each second and the velocity at the end of each second.
[Illustration: FIG. 80.--Apparatus to illustrate uniformly accelerated
motion.]
In Fig. 63, if _OG_ represents the weight of the body or the pull of
gravity, then the line _OR_ will represent the effective component along
the wire, and _OS_ the non-effective component against the wire. Since
the ratio of the height of the plane to its length is as one to sixteen,
then the motion along the wire in Fig. 80 will be one-sixteenth that of
a falling body.
=98. Summary of Results.=--The following table gives the results that
have been obtained with an apparatus arranged as shown above.
In this table, column 2 is the one which contains the results directly
observed by the use of the apparatus. Columns, 3, 4, and 5 are computed
from preceding columns.
(1) (2) (3) (4) (5)
No. of Total Distance Velocity at Acceleration
seconds distance each second end of second each second
moved
Per second Per second
1 30 cm. 30 cm. 60 cm. 60 cm.
2 120 cm. 90 cm. 120 cm. 60 cm.
3 270 cm. 150 cm. 180 cm. 60 cm.
4 480 cm. 210 cm. 240 cm. 60 cm.
Column 5 shows that the acceleration is uniform, or the same each
second. Column 4 shows that the velocity increases with the number of
seconds or that _V_ = _at_. Column 3 shows that the increase in motion
from 1 second to the next is just equal to the acceleration or 60 cm.
This is represented by the following formula: _s_ = 1/2 _a_(2_t_ - 1).
The results of the second column, it may be seen, increase as 1:4:9:16,
while the number of seconds vary as 1:2:3:4. That is, _the total
distance covered is proportional to the square of the number of
seconds_.
This fact expressed as a formula gives: _S_ = 1/2_at_².
Substituting _g_, the symbol for the acceleration of gravity, for _a_ in
the above formulas, we have: (1) _V_ = _gt_, (2) _S_ = 1/2_gt_², (3) _s_
= 1/2_g_(2_t_ - 1).
=99. Laws of Falling Bodies.=--These formulas may be stated as follows
for a body which falls from rest:
1. The velocity of a freely falling body at the end of any second is
equal to 32.16 ft. per sec. or 980 cm. per second multiplied by the
number of the second.
2. The distance passed through by a freely falling body during any
number of seconds is equal to the square of the number of seconds
multiplied by 16.08 ft. or 490 cm.
3. The distance passed through by a freely falling body during any
second is equal to 16.08 feet or 490 cm. multiplied by one less than
twice the number of the second.
Important Topics
1. Falling bodies.
2. Galileo's experiment.
3. Acceleration due to gravity.
4. Laws of falling bodies.
Exercises
1. How far does a body fall during the first second? Account for the
fact that this distance is numerically equal to half the acceleration.
2. (a) What is the velocity of a falling body at the end of the first
second? (b) How far does it fall during the second second? (c) Account
for the difference between these numbers.
3. What is the velocity of a falling body at the end of the fifth
second?
4. How far does a body fall (a) in 5 seconds (b) in 6 seconds (c) during
the sixth second?
5. (a) What is the difference between the average velocity during the
sixth second and the velocity at the beginning of that second?
(b) Is this difference equal to that found in the second problem? Why?
6. A stone dropped from a cliff strikes the foot of it in 5 seconds.
What is the height of the cliff?
7. Why is it that the increased weight of a body when taken to higher
latitudes causes it to fall faster, while at the same place a heavy body
falls no faster than a light one?
8. When a train is leaving a station its acceleration gradually
decreases to zero, although the engine continues to pull. Explain.
9. Would you expect the motion of equally smooth and perfect spheres of
different weight and material to be equally accelerated on the same
inclined plane? Give reason for your answer. Try the experiment.
10. A body is thrown upward with the velocity of 64.32 ft. per sec. How
many seconds will it rise? How far will it rise? How many seconds will
it stay in the air before striking the ground?
11. 32.16 feet = how many centimeters?
12. The acceleration of a freely falling body is constant at any one
place. What does this show about the pull which the earth exerts on the
body?
(7) THE PENDULUM
=100. The Simple Pendulum.=--Any body suspended so as to swing freely to
and fro is a pendulum, as in Fig. 81. A simple pendulum is defined as a
single particle of matter suspended by a cord without weight. It is of
course impossible to construct such a pendulum. A small metal ball
suspended by a thread is approximately a simple pendulum. When allowed
to swing its vibrations are made in equal times. This feature of the
motion of a pendulum was first noticed by Galileo while watching the
slow oscillations of a bronze chandelier suspended in the Cathedral in
Pisa.
[Illustration: FIG. 81--A simple pendulum.]
=101. Definition of Terms.= _The center of suspension_ is the point
about which the pendulum swings. A _single vibration_ is one swing
across the arc. A _complete_ or _double_ vibration is the swing across
the arc and back again. The time required for a double vibration is
called the _period_. The _length_ of a simple pendulum is approximately
the distance from the point of support to the center of the bob.
A _seconds pendulum_ is one making a single vibration per second. Its
length at sea-level, at New York is 99.31 cm. or 39.1 in., at the
equator 39.01 in., at the poles 39.22 in.
A _compound pendulum_ is one having an appreciable portion of its mass
elsewhere than in the small compact body or sphere called a bob. The
ordinary clock pendulum or a meter stick suspended by one end are
examples of compound pendulums.
The _amplitude_ of a vibration is one-half the arc through which it
swings, for example, the arc _DC_ or the angle _DAC_ in Fig. 81.
=102. Laws of the Pendulum.=--The following laws may be stated:
1. The period of a pendulum is not affected by its mass or the material
of which the pendulum is made.
2. For small amplitudes, the period is not affected by the length of the
arc through which it swings.
3. The period is directly proportional to the square root of the length.
Expressed mathematically, _t_/_t´_ = √_l_/√_l´_.
=103. Uses of the Pendulum.=--The chief use of the pendulum is to
regulate motion in clocks. The wheels are kept in motion by a spring or
a weight and the regulation is effected by an escapement (Fig. 82). At
each vibration of the pendulum one tooth of the wheel _D_ slips past the
prong at one end of the escapement _C_, at the same time giving a slight
push to the escapement. This push transmitted to the pendulum keeps it
in motion. In this way, the motion of the wheel work and the hands is
controlled. Another use of the pendulum is in finding the acceleration
of gravity, by using the formula, _t_ = π√(_l_/_g_), in which _t_ is the
time in seconds of a single vibration and _l_ the length of the
pendulum. If, for example, the length of the seconds pendulum is 99.31
cm., then 1 = π√(99.31/_g_); squaring both sides of the equation, we
have 1² = π²(99.31/_g_), or _g_ = π² × 99.31/1² = 980.1 cm. per sec.,
per sec. From this it follows that, since the force of gravity depends
upon the distance from the center of the earth, the pendulum may be used
to determine the elevation of a place above sea level and also the shape
of the earth.
Important Topics
1. Simple pendulum.
2. Definitions of terms used.
3. Laws of the pendulum.
4. Uses of the pendulum.
Exercises
1. What is the usual shape of the bob of a clock pendulum? Why is this
shape used instead of a sphere?
2. Removing the bob from a clock pendulum has what effect on its motion?
Also on the motion of the hands?
3. How does the expansion of the rod of a pendulum in summer and its
contraction in winter affect the keeping of time by a clock? How can
this be corrected?
4. Master clocks that control the time of a railway system have a cup of
mercury for a bob. This automatically keeps the same rate of vibration
through any changes of temperature. How?
5. How will the length of a seconds pendulum at Denver, 1 mile above
sea-level, compare with one at New York? Why?
[Illustration: FIG. 82--Escapement and pendulum of a clock.]
6. What is the period of a pendulum 9 in. long? _Note._ In problems
involving the use of the third law, use the length of a seconds pendulum
for _l_, and call its period 1.
7. A swing is 20 ft. high, find the time required for one swing across
the arc.
8. A pendulum is 60 cm. long. What is its period?
9. If in a gymnasium a pupil takes 3 sec. to swing once across while
hanging from a ring, how long a pendulum is formed?
10. A clock pendulum makes four vibrations a second, what is its
length?
Review Outline: Force and Motion
Force; definition, elements, how measured, units, dyne.
Graphic Representation; typical examples of finding a component, a
resultant, or an equilibrant.
Motion; Laws of motion (3), inertia, curvilinear motion, centrifugal
force, momentum, (_M = mv_), reaction, stress and strain.
Moment of Force; parallel forces, couple, effective and non-effective
component.
Gravitation; law; gravity, center of; weight. Equilibrium 3 forms;
stability, how increased.
Falling Bodies; velocity, acceleration, "g," Laws; _V_ = _gt_, _S_ =
(1/2)_gt_² - _s_ = (1/2)_g_(2_t_ - 1).
Pendulum; simple, seconds, laws (3), _t_ = π√(_l_/_g_).