that humid air at a given temperature and pressure is lighter than dry air at the same temperature and pressure.
Water Content of the Atmosphere In the troposphere the air is seldom completely dry. It contains water vapor in either of two forms: (1) Fog or (2) water vapor. Fog consists of minute droplets of water held in suspension by the air. Clouds are composed of fog. The height to which some clouds extend is a good indication of the presence of water in the atmosphere almost up to the stratosphere.
As a result of evaporation, the atmosphere always contains some moisture in the form of water vapor. The moisture in the air is called the humidity of the air. Moisture does not consist of tiny particles of liquid held in suspension in the air as in the case of fog, but is an invisible vapor truly as gaseous as the air with which it mixes.
Fog and humidity both affect the performance of an aircraft. In flight, at cruising power, the in use. Some merely show the expected reduction in power due to humidity; others show the boost in manifold pressure necessary to restore full takeoff power.
The effect of fog on the performance of an engine is very noticeable, particularly on engines with high compression ratios. Normally, some detonation will occur during acceleration, due to the high BMEP (Brake Mean Effective Pressures) developed. However, on a foggy day it is difficult to cause detonation to occur. The explanation of this lies in the fact that fog consists of unvaporized particles of water. When these particles enter the cylinders, they absorb a tremendous amount of heat energy in the process of vaporizing. The temperature is thus lowered, and the decrease is Fog will generally cause a decrease in horsepower output. However with a supercharged engine, it will be possible to use higher manifold pressures without danger of detonation.
effects are small and receive no consideration.
During takeoff, however, humidity has important effects. Two things are done to compensate for the effects of humidity on takeoff performance.
Since humid air is less dense than dry air, the allowable takeoff gross weight of an aircraft is generally reduced for operation in areas that are consistently humid. Secondly, because the power output of reciprocating engines is decreased by humidity, the manifold pressure may have to be increased above that recommended for takeoff in dry air in order to obtain the same power output.
Engine power output is calculated on dry air.
Since water vapor is incombustible, its pressure in the atmosphere is a total loss as far as contributing to power output. The mixture of water vapor and air is drawn through the carburetor and fuel is metered into it as though it were all air. This mixture of water vapor, air, and fuel enters the combustion chamber where it is ignited Since the water vapor will not burn, the effective fuel/air ratio is enriched and the engine operates as though it were on an excessively rich mixture.
The resulting horsepower loss under humid conditions can therefore be attributed to the loss in volumetric efficiency due to displaced air, and the incomplete combustion due to excessively rich fuel/air mixture.
The reduction in power that can be expected from humidity is usually given in charts in the flight manual. There are several types of charts Absolute Humidity Absolute humidity is the actual amount of the water vapor in a mixture of air and water. It is sometimes expressed in g./cu.m. (grams per cubic meter), sometimes in lbs./cu. ft. The amount of water vapor that can be present in the air is dependent upon the temperature and pressure.
The higher the temperatures, the more water vapor the air is capable of holding, assuming constant pressure. When air has all the water vapor it can hold at the prevailing temperature and pressure, it is said to be saturated.
Relative Humidity Relative humidity is the ratio of the amount of water vapor actually present in the atmosphere to the amount that would be present if the air were saturated at the prevailing temperature and pressure. This ratio is usually multiplied by 100 and expressed as a percentage. Suppose, for example, that a weather report includes the information that the temperature is 75° F. and the relative humidity is 56 percent. This indicates that the air holds 56 percent of the water vapor required to saturate it at 75° F. If the temperature drops and the absolute humidity remains constant, the relative humidity will increase. This is because less water vapor is required to saturate the air at the lower temperature.
Sling type Continuous reading type FIGURE 7-20. Wet-bulb hygrometers.
Dew Point The dew point is the temperature to which humid air must be cooled at constant pressure to become saturated. If the temperature drops below the dew point, condensation occurs. People who wear glasses have had the experience of going from cold outside air into a warm room and having moisture collect quickly on their glasses.
This happened because the glasses were below the dew point temperature of the air in the room.
The air immediately in contact with the glasses was cooled below its dew point temperature, and some of the water vapor was condensed out. This principle is applied in determining the dew point.
A vessel is cooled until water vapor begins to condense on its surface. The temperature at which this occurs is the dew point.
Vapor Pressure Vapor pressure is the portion of atmospheric pressure that is exerted by the moisture in the air (expressed in tenths of an inch Hg). The dew point for a given condition depends on the amount of water pressure present; thus a direct relationship exists between the vapor pressure and the dew point.
Wet- and Dry-Bulb Temperatures Vapor pressure and humidity may be determined from charts based on the wet- and dry-bulb temperatures (figure 7-20). The dry-bulb temperature is obtained from an ordinary thermometer.
The wet-bulb temperature is obtained from a thermometer which has its bulb covered with a thin piece of wet cloth.
Because of moisture evaporation from the wet cloth, the wet bulb will read lower than the dry bulb. The more rapid the evaporation, the greater will be the difference in readings. The rate of evaporation is dependent upon the degree of saturation of the air. In using the wet-bulb thermometer, it must be moved through the air at a rate of about 1,200 f.p.m. to give a worthwhile reading. This is accomplished by mounting both the wet- and dry-bulb thermometers on a frame which can be hand rotated around a pivot and the desirable rate of 1,200 f.p.m.
attained.
If the air is saturated, no evaporation will take place and the wet- and dry-bulb temperatures will be the same. Thus, these two temperatures coincide at the dew point.
Physical Laws Related to the Atmosphere Although air is composed of various gases and must be treated as a mixture for certain purposes, it is treated as a uniform gas in aerodynamic calculations. Air is a fluid since it has the fluid property to flow, and it is also a gas, since its density is readily varied.
As is usual in engineering work, certain simplifying assumptions are made. One standard assumption is that in dry air there is no water vapor present. Ordinary flight-takeoff charts may be corrected for vapor pressure, but subsonic 240
flight does not consider vapor pressure as an appreciable factor. Another standard assumption is that friction or "viscosity effect" may be neglected when dealing with airflow. The air is then considered to be a perfect fluid. However, some exceptions must be made, particularly in the case of thin boundary layers of slow moving air directly adjacent to a moving body.
The Kinetic Theory of Gases Applied to Air The kinetic theory states that a gas is composed of small, distinct particles called molecules. The size of the molecules is small compared to the average distance between them. Further, the molecules are moving about at a high rate of speed in random directions so that they are constantly colliding with one another and with the walls of any container that may surround them.
The pressure produced by a gas is the result of these continuous impacts against a surface, and since the impacts are essentially infinite in number, a steady pressure is effected.
Just as pressure is produced by molecular impact against a surface, it is also transmitted by molecular impact. Assuming that molecules are perfectly elastic (that no friction exists between them, a pressure wave once started will continue indefinitely. For most purposes this assumption is adequate; however, it is not completely correct.
For instance, sound represents a series of weak pressure waves to which the ear is sensitive. If the energy that the sound represents were not lost, the sound would continue indefinitely. It follows then that this imperfect elasticity must in some way be associated with fluid friction or viscosity, since the presence of viscosity is also a source of energy loss.
On the basis of the kinetic theory, pressure may be increased in two ways: First, by increasing the number of molecules in a given space, which is the same as increasing the density; and secondly, by increasing the speed of the molecules, which can be done by increasing the temperature, since the temperature increase produces an increase in the molecular speed.
Analysis of the kinetic theory leads to one definite relationship between the temperature, pressure, and density of a gas when the gas is subjected to a given set of conditions. This relationship is known as the equation of state.
Equation of State Provided that the temperature and pressure of a gas are not excessively different from those normally experienced on the earth's surface, the following equation holds true:
PV = RT where: P = pressure in lbs./sq. ft.
V - specific volume.
R = a constant for a given gas (for air R = 53.345).
T = absolute temperature (Rankine = °F. + 459.4).
If the temperature and pressure are such that the gas becomes a liquid, or if the pressure falls to such a value that uniform pressure no longer exists, the equation will be invalid. In practical aeronautical work, these extremes are only encountered in a supersonic wind tunnel or in the outer portions of the atmosphere. This formula must be further clarified for practical engineering, by the introduction of air density.
Standard Atmosphere If the performance of an aircraft is computed, either through flight tests or wind tunnel tests, some standard reference condition must be determined first in order to compare results with those of similar tests. The conditions in the atmosphere vary continuously, and it is generally not possible to obtain exactly the same set of conditions on two different days or even on two successive flights. Accordingly, there must be set up a group of standard conditions that may be used arbitrarily for reference. The set of standard conditions presently used in the United States is known as the U.S. Standard Atmosphere.
The standard atmosphere approximates the average conditions existing at 40° latitude, and is determined on the basis of the following assumptions. The standard sea level conditions are:
Pressure at 0 altitude (Po) = 29.92 inches of mercury.
Temperature at 0 altitude (T%) = 15° C. = 59° F.
Gravity at 0 altitude (go) = 32.174 ft./sec.2 The U.S. Standard Atmosphere is in agreement with the International Civil Aviation Organization (ICAO) Standard Atmosphere over their common altitude range. The ICAO Standard Atmosphere 241
has been adopted as standard by most of the principal nations of the world.
Variations from Standard Days As may be expected, the temperature, pressure, density, and water vapor content of the air varies considerably in the troposphere. The temperature at 40° latitude may range from 50° C. at low altitudes during the summer to -70° C. at high altitudes during the winter. As previously stated, temperature usually decreases with an increase in altitude. Exceptions to this rule occur when cooler air is trapped near the earth by a warmer layer.
This is called a temperature inversion, commonly associated with frontal movement of air masses.
Pressure likewise varies at any given point in the atmosphere. On a standard day, at sea level, pressure will be 29.92 in Hg. On nonstandard days, pressure at sea level will vary considerably above or below this figure.
Density of the air is determined by the pressure and temperature acting upon it. Since the atmosphere can never be assumed to be "standard," a convenient method of calculating density has been devised. Since air pressure is measured in inconvenient terms, it is expedient to utilize the aneroid altimeter as a pressure gage and refer to the term "pressure altitude" instead of atmospheric pressure.
Pressure Altitude Pressure altitude is the altitude in the standard atmosphere corresponding to a particular value of air pressure. The aircraft altimeter is essentially a sensitive barometer calibrated to indicate altitude in the standard atmosphere.
With the altimeter of an aircraft set at 29.92 in.
Hg, the dial will indicate the number of feet above or below a level where 29.92 in. Hg exists, not necessarily above or below sea level, unless standard day conditions exist. In general, the altimeter will indicate the altitude at which the existing pressure would be considered standard pressure. The symbol H, is used to indicate pressure altitude.
BERNOULLI'S PRINCIPLE General In the earlier discussion of fluids, Bernoulli's principle was introduced to explain the relationship between velocity and pressure of a liquid flowing through a venturi. Since Bernoulli's principle applies to fluids, which by definition include both gases and liquids, its application to gases (air) is included at this point to explain the relationship between air velocity and pressure on the surfaces of an airfoil.
How an Aircraft Wing Reacts with the Atmosphere An airfoil is any surface designed to obtain a reaction from the air through which it moves.
Wings, ailerons, elevators, stabilizers, propeller blades, and helicopter rotors are all airfoils.
The reaction for which wings are designed is called lift. A wing produces lift because of a pressure difference; and the greater this difference, the more lift developed. If the air pressure above a wing is the same as that below it, there is no lift. But if the pressure above a wing is reduced and the pressure below increased, then lift is produced. The strong air pressure below the wing moves the wing upward against the weaker pressure above the wing. But what causes these unequal pressures?
An examination of the shape of an aircraft wing discloses that it has been designed to create a pressure difference. If a wing is cut from leading edge to trailing edge, the end view of the cut is a profile section similar to the one shown in figure
7-21. The forward part of the airfoil profile is rounded and is called the leading edge. The aft part is narrow and tapered and is called the trailing edge. A reference line often used in discussing airfoils is the chord, an imaginary straight line joining the extremities of the leading and trailing edges. The curved upper surface of the airfoil is called the camber. The lower surface is normally straight, or only slightly curved.
An airfoil is very similar in shape to half a venturi section. In A of figure 7-22, the throat, or restricted portion of a venturi is illustrated.
The airflow through the venturi is indicated by flow lines. In B of figure 7-22, one-half of a Camber Leading _ edge Trailing Chord line. edger FIGURE 7-21. Cross-sectional view of an airfoil.
B FIGURE 7-22. Airflow in venturi sections.
venturi restriction is shown, together with the airflow over its curved surface. Note that this portion of a venturi has the same profile as that of an airfoil.
To understand how lift is produced by an aircraft's wings, Bernoulli's principle is applied to an airfoil. This principle reveals that the pressure of fluid (liquid or gas) decreases at points where the speed of the fluid increases. In other words, high speed is associated with low pressure, and low speed with high pressure. The wing, or airfoil, of an aircraft is designed to increase the velocity of the airflow above its surface, thereby decreasing pressure above the airfoil. Simultaneously, the impact of the air on the lower surface of the airfoil increases the pressure below. This combination of pressure decrease above the airfoil and increase below the airfoil produces lift.
MACHINES General Ordinarily, a machine is thought of as a complex device, such as an internal-combustion engine or a typewriter. These are machines, but so is a hammer, a screwdriver, or a wheel. A machine is any device with which work may be accomplished.
Machines are used to transform energy, as in the case of a generator transforming mechanical energy into electrical energy. Machines are used to transfer energy from one place to another, as in the examples of the connecting rods, crankshaft, and reduction gears transferring energy from an aircraft's engine to its propeller.
Another use of machines is to multiply force;
for example, a system of pulleys may be used to lift a heavy load. The pulley system enables the load to be raised by exerting a force which is smaller than the weight of the load.
Machines are also used to multiply speed. A good example is the bicycle, by which speed can be gained by exerting a greater force.
R FIGURE 7-23. A simple lever.
Finally, machines can be used to change the direction of a force. An example of this use is the flag hoist. A downward force on one side of the rope exerts an upward force on the other side, raising the flag toward the top of the pole.
There are only six simple machines. They are the lever, the pulley, the wheel and axle, the inclined plane, the screw, and the gear. However, physicists recognize only two basic principles in machines; namely, the lever and the inclined plane.
The wheel and axle, the block and tackle, and gears may be considered levers. The wedge and the screw use the principle of the inclined plane.
An understanding of the principles of simple machines provides a necessary foundation for the study of compound machines, which are combinations of two or more simple machines.
The Lever The simplest machine, and perhaps the most familiar one, is the lever. A seasaw is a familiar example of a lever in which one weight balances the other.
There are three basic parts in all levers; namely, the fulcrum "F," a force or effort "E," and a resistance "R." Shown in figure 7-23 are the pivotal point "F" (fulcrum); the effort "E," which is applied at a distance "A" from the fulcrum;
and a resistance "R," which acts at a distance "a" from the fulcrum. Distances " 4" and "a" are the lever arms.
Classes of Levers The three classes of levers are illustrated in figure 7-24. The location of the fulcrum (the fixed or pivot point) with relation to the resistance (or weight) and the effort determines the lever class.
Fulcrum this case, as in A of figure 7-24, the force is applied on one side of the fulcrum, and the re- Effort Weignt sistance to be overcome is applied to the opposite A. First-class lever side; hence, this is a first-class lever. Crowbars, shears, and pliers are common examples of this Effort class of lever.
Fulcrum Weight B. Second-class lever Effort Fulcrum Weight C. Third-class lever FIGURE 7-24. Three classes of levers.
First-Class Levers In the first-class lever (A of figure 7-24), the fulcrum is located between the effort and the re sistance. As mentioned earlier, the seesaw is a good example of the first-class lever. The amount of weight and the distance from the fulcrum can be varied to suit the need. Another good example Second-Class Levers The second-class lever (B of figure 7-24) has the fulcrum at one end; the effort is applied at the other end. The resistance is somewhere between these points. The wheelbarrow in figure 7-26 is a good example of a second-class lever.
Both first- and second-class levers are commonly used to help in overcoming big resistances with a relatively small effort.
Third-Class Levers There are occasions when it is desirable to speed up the movement of the resistance even though a large amount of effort must be used.
Levers that help accomplish this are third-class levers. As shown in C of figure 7-24, the fulcrum is at one end of the lever and the weight or resistance to be overcome is at the other end, with the effort applied at some point between.
FIGURE 7-26. A second-class lever.
distance "e," the resistance "R" moves a greater distance "r." The speed of "R" must be greater than that of "E," since "R" covers a greater distance in the same length of time.
The human arm (figure 7-28) is a third-class lever. It is this lever action that makes possible the quick flexing of the arms. Note that the elbow is the fulcrum. The biceps muscle, which is tied onto the forearm below the elbow, applies the effort; and the hand, is the resistance. Third-class levers should be used to gain speed, rather than to move heavy loads.
The forces required to operate machines, as well as the forces they will exert, can be easily determined. Consider the iron bar used as a firstclass lever in figure 7-29. The bar is 9 feet long and is used to raise a 300-pound weight. Assume that a maximum of 100 pounds is available to lift the weight. If a fulcrum "p" is placed 2 feet from the center of the weight, a 6-foot length of the bar becomes the effort arm. This 6-foot length R FIGURE 7-28. The arm is a third-class lever.
is three times as long as the distance from the fulcrum to the center of the weight. With a 100- pound effort at "E", the 300-pound weight can be lifted, since the length of the effort arm has multiplied the effort three times. This is an example of the direct relationship between lengths of lever arms and the forces acting on the arms.
This relationship can be stated in general terms:
The length of the effort arm is the same number of times greater than the length of the resistance arm as the resistance to be overcome is greater than the effort that must be applied.
The mathematical equation for this relation- Effort (E) Resistance arm 300 1bs. (1) gr. wt. - Effort arm (L) Hesistance (R) Fulcrum (F) FIGURE 7-29. Computing the forces in a FIGURE 7-27. A third-class lever. first-class lever.
Effort The same general formula applies for secondclass levers; but it is important to measure the proper lengths of the effort arm and the resistance Resistance arm. Referring to figure 7-26, the length of the wheelbarrow handles from the axle of the wheel (which is the fulcrum) to the grip is 4 feet. This
- Fulcrum L = 5"- load of sand is 1 foot from the axle; thus, the is an effort arm 4 feet in length. The center of the length of the resistance arm is 1 foot.
By substituting in the formula, = E R FIGURE 7-30. A first-class lever problem. E = 50 ib.
ship is:
R E where: L = length of effort arm.
1 = length of resistance arm.
R = resistance weight or force.
E = effort force.
Remember that all distances must be in the same units, and all forces must also be in the same units.
In figure 7-30 another first-class lever problem is illustrated: To pry up the lid of the paint can with a 6-inch bar when the average force holding the lid is assumed to be 50 pounds. If the distance from the edge of the can to the edge of the cover is 1 inch, what force must be applied to the end of the bar?
A third-class lever problem is illustrated in figure 7-28. With one hand, a weight of 10 pounds is to be lifted. If the biceps muscle is attached to the forearm 1 inch below the elbow, and the distance from the elbow to the palm of the hand is 18 inches, what pull must the muscle exert in order to hold the weight and flex the arm at the elbow? By substituting in the formula, L R it becomes 18 _ 10 and E = 18 X 10 = 180 lb.
The muscle must exert a 180-pound pull to hold up the 10-pound weight. This illustrates that the biceps muscle is poorly arranged for lifting or pulling. It also illustrates that third-class levers should be used primarily to speed up motion of a resistance.
According to the formula:
1 =F R Mechanical Advantage of Levers 1 Levers may provide mechanical advantages, Here, L = 5 inches; 1 = 1 inch; R = 50 pounds, since they can be applied in such manner that and E is unknown. Substitute the numbers in they can magnify an applied force. This is true of first- and second-class levers. The third-class their proper places; then lever provides what is called a fractional dis- 5 50 advantage, i.e., one in which a greater force is required than the force of the load lifted.
In the problem involving the wheelbarrow (figand E = 50X1 ure 7-26), a 50-pound pull overcomes the 200- 5 = 10 pounds. pound weight of the sand. In this case, the effort was magnified four times. Thus the mechanical A force of 10 pounds is required. advantage gained by using the wheelbarrow is 4.
FIGURE 7-31. An inclined plane.
Expressing the same idea in mathematical terms, Resistance Mechanical Advantage = Effort or MA = E R Thus, in the case of the wheelbarrow, M.1 - 50 = 4.
This rule applies to all machines.
Mechanical advantage of levers may also be found by dividing the length of the effort arm "4" by the length of the resistance arm "a." Stated as a formula, this reads:
Effort Arm Mechanical Advantage = Resistance Arm or MA = 1 a How does this apply to third-class levers? If a muscle pulls with a force of 1,800 pounds in order to lift a 100-pound projectile, a mechanical advantage of 101,800 or 18 is obtained. This is a fractional disadvantage, since it is less than l.
The Inclined Plane The inclined plane is a simple machine that facilitates the raising or lowering of heavy objects by application of a small force over a relatively long distance. Some familiar examples of the inclined plane are mountain highways and cattle loading ramps.
The inclined plane permits a large resistance to be overcome by application of a small force through a longer distance than the load is to be raised. In figure 7-31, a 300-pound barrel is being rolled up a ramp to the bed of a truck, 3 feet above the sidewalk. The ramp is 9 feet long.
Without the ramp, a force of 300 pounds, applied straight up through the 3-foot distance, would be required to load the barrel. With the ramp, a force can be applied over the entire 9 feet of the ramp as the barrel is rolled slowly up to a height of 3 feet. It can be determined by observation that a force of only three-ninths of 300, or 100 pounds, will be required to raise the barrel by using an inclined plane. This can also be determined mathematically, using the formula, R where:
L = length of the ramp, measured along the slope.
1 = height of the ramp.
R = weight of object to be raised, or lowered.
E = force required to raise or lower object.
In this case, L = 9 ft.; 1 = 3 ft.; and R = 300 lb.
Substituting these values in the formula, 9E = 900 E = 100 pounds.
Since the ramp is three times as long as its height, the mechanical advantage is three. The theoretical mechanical advantage is found by dividing the total distance through which the effort is exerted by the vertical distance through which the load is raised or lowered.
The Wedge The wedge is a special application of the inclined plane. The blades of knives, axes, hatchets, and 247
100# 600# 600# FIGURE 7-32. A wedge.
chisels act as wedges when they are forced into a piece of wood. The wedge is two inclined planes, set base-to-base. By driving the wedge full-length into the material to be cut or split, the material is forced apart a distance equal to the width of the broad end of the wedge (see figure 7-32).
Long, slim wedges have high mechanical advantages. For example, the wedge in figure 7-32 has a mechanical advantage of 6. The greatest value of wedges is found in situations where other simple machines cannot be used. For example, imagine trying to pull a log apart with a system of pulleys.
The Pulley Pulleys are simple machines in the form of a wheel mounted on a fixed axis and supported by a frame. The wheel, or disk, is normally grooved to accommodate a rope. The wheel is sometimes referred to as a "sheave" (sometimes "sheaf").
The frame that supports the wheel is called a block. A block and tackle consists of a pair of blocks. Each block contains one or more pulleys and a rope connecting the pulley (s) of each block.
Single Fixed Pulley A single fixed pulley is really a first-class lever with equal arms. The arms "EF" and "FR" in figure 7-33 are equal; hence the mechanical advantage is one. Thus, the force of the pull on the rope must be equal to the weight of the object being lifted. The only advantage of a single fixed pulley is to change the direction of the force, or pull on the rope.
A single pulley can be used to magnify the force exerted. In figure 7-34 the pulley is not 200 Ibs FIGURE 7-33. A single fixed pulley. FIGURE 7-34. A single movable pulley.
E FIGURE 7-35. A single movable pulley as a second-class lever.
fixed, and the rope is doubled because it supports a 200-pound weight. Used in this manner, a single block and tackle can lift the 200-pound weight with a 100-pound pull, since each half of the rope (tackle) carries one-half the total load. The mechanical advantage is 2, which can be verified by using the formula:
MA - · - 100 = 2.
The single movable pulley used in the manner shown in figure 7-34 is a second-class lever. To see this refer to figure 7-35. The effort "E" acts upward on the arm "EF," which is the diameter of the pulley. The resistance "R" acts downward on the arm "FR," which is the radius of the pulley. Since the diameter is twice the radius, the mechanical advantage is 2.
However, when the effort at "E" moves up 2 feet, the load at "R" is raised only 1 foot. This is true of all systems of block and tackle, for if a mechanical advantage is obtained, the length of rope passed through the hands is greater than the distance that the load is raised.
The mechanical advantage of a pulley system is found by measuring the resistance and the effort and dividing the amount of resistance by the effort. A shorthand method often used is simply to count the number of rope strands that move or support the movable block.
WORK, POWER, AND ENERGY Work The study of machines, both simple and complex, is in one sense a study of the energy of mechanical work. This is true because all machines transfer input energy, or the work done on the machine to output energy, or the work done by the machine.
Work, in the mechanical sense of the term, is done when a resistance is overcome by a force acting through a measurable distance. Two factors are involved: (1) Force and (2) movement through a distance. As an example, suppose a small aircraft is stuck in the snow. Two men push against it for a period of time, but the aircraft does not move. According to the technical definition, no work was done in pushing against the aircraft.
By definition, work is accomplished only when an object is displaced some distance against a resistive force.
In equation form, this relationship is, Work=Force (F) x distance (d).
The physicist defines work as "work is force times displacement. Work done by a force acting as a body is equal to the magnitude of the force multiplied by the distance through which the force In the metric system, the unit of work is the joule, where one joule is the amount of work done by a force of one newton when it acts through a distance of one meter. That is, 1 joule = 1 newton-m Hence we can write the definition in the form W (joules) =F (newtons) x d (meters) If we push a box for 8 m across a floor with a force of 100 newtons, the work we perform is W= Fd =100 newtons × 8 m =800 joules 249
How much work is done in raising a 500-kg (kilogram) elevator cab from the ground floor of a building to its tenth floor, 30 m (meters) higher?
We note that the force needed is equal to the weight of the cab, which is mg.
In the metric system, mass rather than weight is normally specified. To find the weight in newtons (the metric unit of force) of something whose mass in kilograms is known, we simply turn to f=mg and set G=9.8 m/SEC?
F (newtons) =M (kilograms) x G (9.8 m/sec?)
W (joules) =M (kilograms) x G (9.8 m/sec?) x D (meters) V=fd=mgd=500 kg x 9.8 m/Sec? × 30m = 147,000 joules =1.47 x 10s joules Force Parallel To Displacement If force is expressed in pounds and distances in feet, work will be expressed in ft. Ibs. (foot-pounds).
Example: How much work is accomplished in lifting a 40 pound weight to a vertical height of 25 feet?
When a force and the distance through which it acts are parallel, the work done is equal to the product of F and d.
W = F X d When they are not in the same direetion, the work done is equal to the product of d and the component of F in the direction of d, namely (F cos 8) X d.
Component of F in direction of displacement FenSe = 1,000 ft.-lb.
Example: How much work is accomplished in pushing a small aircraft into a hangar a distance of 115 feet if a force of 75 pounds is required to keep it moving?
W=Fd = 75 × 115 ==8,625 ft.-lb.
Force Not Parallel To Displacement In this equation, F is assumed to be in the same direction as d. If it is not, for example the case of a body pulling a wagon with a rope not parallel to the ground, we must use F for the component of the applied force that acts in the direction of the motion, figure 7-37(A).
The component of a force in the direction of a displacement d is:
F cos 8 where & is the angle between F and d. Hence the most general equation for work is W= Fd cos 8 When F and d are parallel, 0=0 and cos 0=1, so that Fd cos 0 reduces to just Fd. When F and d are perpendicular, 0=90° and cos 0=0, so that no work is done. A force that is perpendicular to the motion of an object can do no work upon it.
Thus gravity, which results in a downward force on everything near the earth, does no work on objects moving horizontally along the earth's surface. However, if we drop an object, as it falls to the ground work is definitely done upon it.
W' = (F cOS A) X d FICURE 7-36. Direction of work.
Friction In calculating work done, the actual resistance overcome is measured. This is not necessarily the weight of the object being moved. This point can be illustrated by referring to figure 7-37. A 900- pound load is being pulled a distance of 200 feet.
This does not mean that the work done (force x distance) is 180,000 foot-pounds (900 pounds x 200 feet). This is because the man pulling the load is not working against the total weight of the load, but rather against the rolling friction of the cart, which may be no more than 90 pounds.
Friction is one of the most important aspects of life. Without friction it would be impossible to walk. One would have to shove oneself from place to place, and would have to bump against some obstacle to stop at a destination. Yet friction is a liability as well as an asset, and requires consideration when dealing with any moving mechanism.
In experiments relating to friction, measurement of the applied forces reveals that there are three kinds of friction. One force is required to start a body moving, while another is required to keep the body moving at constant speed. Also, after a body is once in motion, a definitely larger force is required to keep it sliding than to keep it rolling.