Physics Part 15

[Ill.u.s.tration: 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 quant.i.ty 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: _AB + BD = AD_. Why?

and _AD_ = v_([line]AB + [line]BD)._

[Ill.u.s.tration: FIG. 56.--Graphic representation of forces acting along the same or parallel lines.]

[Ill.u.s.tration: 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 cla.s.sified 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 pa.s.sed 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.

[Ill.u.s.tration: 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 ma.s.ses of the objects considered. Thus, one may speak of a pound-ma.s.s 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 ma.s.s, 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 ma.s.ses. 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-ma.s.s_ and _gram-weight_. The ma.s.s 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. Cla.s.sification 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...o...b..t, and in following the sun in its motion through s.p.a.ce. Motions are cla.s.sified 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_.