### FORCE AND MOTION

Chapter V. 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[Pg 80] 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. 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[Pg 81] and (c) its magnitude, or 5 lbs. These three elements may be represented by a line. Thus in Fig. 56a, 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. 56b. 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. 56c), 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,[Pg 82] 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). Fig. 56.—Graphic representation of forces acting along the same or parallel lines. 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[Pg 83] 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. 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[Pg 84] 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.
[Pg 85]
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[Pg 86] 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[Pg 87] 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.

### 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[Pg 88] 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.) 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. 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. 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.
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[Pg 92] 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.) 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.
[Pg 93]
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[Pg 94] 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.
[Pg 95]
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?
[Pg 96]
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[Pg 97] AB is used in drawing the sled forward. Its effectiveness is indicated by the relative size of the component AD compared to AB. 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? Fig. 63.—The effective component is OR.
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[Pg 98] 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. 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?
[Pg 99]
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. 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[Pg 100] 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. 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 . 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 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.
[Pg 101]
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. 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.
[Pg 102]

#### 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? Fig. 69.—Forces acting upon a stretched rope.
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.?
[Pg 103]
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. 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:
[Pg 104]
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[Pg 105] 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[Pg 106] 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. 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.[Pg 107] Other examples are a rocking chair, and the combination shown in Fig. 74. 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). Fig. 74—An example of stable equilibrium. Why? 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).
[Pg 108] 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.)? 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.
[Pg 109]

#### 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[Pg 110] 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. Fig. 78.—Bodies fall alike in a vacuum.
Fig. 79.—Leaning tower of Pisa.
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.
[Pg 111]
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.
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[Pg 112] 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. 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.
[Pg 113]
 (1) No. of seconds (2) Total distance moved (3) Distance each second (4) Velocity at end of second (5) Acceleration each second 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(2t - 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 = 1/2at2.
Substituting g, the symbol for the acceleration of gravity, for a in the above formulas, we have: (1) V = gt, (2) S = 1/2gt2, (3) s = 1/2g(2t - 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.
[Pg 114]

#### 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?
[Pg 115]

### (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. 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[Pg 116] 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/ = √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 = [Pg 117]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? 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?
[Pg 118]

#### 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)gt2 - s = (1/2)g(2t - 1).
Pendulum; simple, seconds, laws (3), t = π√(l/g).