Physics Part 18

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?

[Ill.u.s.tration: FIG. 69.--Forces acting upon a stretched rope.]

[Ill.u.s.tration: 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.

[Ill.u.s.tration: 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...o...b..t 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 ma.s.ses and inversely proportional to the square of the distance between them._

The law may be separated into two parts, one referring to the ma.s.ses 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 ill.u.s.trate 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 ill.u.s.tration 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 pa.s.sing 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 pa.s.sing 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 ma.s.s_ of the body.

The earth's attraction for a body is considered for the sake of simplicity, not as a mult.i.tude 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_.

[Ill.u.s.tration: 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 pa.s.ses 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.

[Ill.u.s.tration: 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_).

[Ill.u.s.tration: FIG. 74--An example of stable equilibrium. Why?]

[Ill.u.s.tration: 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).

[Ill.u.s.tration: 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 pa.s.s 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.)?

[Ill.u.s.tration: 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?