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 ill.u.s.trate: 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_.

[Ill.u.s.tration: 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?

[Ill.u.s.tration: FIG. 63.--The effective component is _OR_.]

[Ill.u.s.tration: 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 pa.s.sing over the Detroit river.

[Ill.u.s.tration: 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 quant.i.ty of motion is called _momentum_ and is measured by the product of the _ma.s.s times the velocity_. In the study of _parallel forces_, especially such as tend to produce _rotation_, we consider a similar quant.i.ty. 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 quant.i.ties; _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_.

[Ill.u.s.tration: FIG. 66.--The moments about _S_ are equal.]

_To ill.u.s.trate:_ 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._

[Ill.u.s.tration: FIG. 67.--Law of parallel forces ill.u.s.trated.]

=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 ill.u.s.trate 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 ill.u.s.tration 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.

[Ill.u.s.tration: 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 compa.s.s 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.