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94. Acceleration of a freely falling body. If in the above experiment the slope of the plane be made steeper, the results will obviously be precisely the same, except that the acceleration has a larger value. If the board is tilted until it becomes vertical, the body becomes a freely falling body (Fig. 79). In this case the distance traversed the first second is found to be 490 centimeters, or 16.08 feet. Hence the acceleration, expressed in centimeters, is 980; in feet, 32.16. This acceleration of free fall, called the acceleration of gravity, is usually denoted by the letter g. For freely falling bodies, then, the three formulas of the preceding paragraph become
To illustrate the use of these formulas, suppose we wish to know with what velocity a body will hit the earth if it falls from a height of 200 meters, or 20,000 centimeters. From (6) we get
v = √2 × 980 × 20,000
95. Height of ascent. If we wish to find the height S to which a body projected vertically upward will rise, we reflect that the time of ascent must be the initial velocity divided by the upward velocity which the body loses per
6261 cm. per second.
second, that is, t ==; and the height reached
FIG. 79. A freely falling body
must be this multiplied by the average velocity
v = √2 gS.
Si (7) is the same as (6), we learn that in a vacuum the speed with which a body must be projected upward to rise to a given height is the same as the speed which it acquires in falling from the same height.
96. Path of a projectile. Imagine a projectile
to be shot along the line ab (Fig. 80). If it
16.08 ft., 64.32 ft., 144.72 ft., etc. Hence it would follow the path indicated by the dotted curve (a parabola). But because of air resistance the height of flight and range are diminished, and the general shape of the trajectory is similar to the continuous curved line.
; that is,
FIG. 80. Path of a projectile
97. The airplane. The principles underlying stability, as well as those having to do with the resolution of forces, are well illustrated by the modern airplane, which grew out of a study of the laws of air resistance and the properties of gliders.
When a plate of area A moves in still air in a direction perpendicular to its plane, with a velocity V (see Fig. 81, (1)), the air resistance R is found by experiment to be given by the equation
R = KAV 2,
(8) where R is the force in kilograms, A the area in square meters, V the speed in meters per second, and K a constant which has the value .08. Thus, when an automobile is going 40 miles
per hour (18 meters per second), the force of the air against .5 square meter of wind-shield is .08 x .5 x (18)2 = 13 kg. When the plate moves so that the direction of its motion makes a small angle i (between 0° and 10°) (Fig. 81, (2)) with its plane, the air resistance R is perpendicular to the plate and is given by the empirical formula
R = KAV 2i,
where R, A, and V have the same significance as above, i is the angle in degrees, and k is very near to .005.
As i, which is called the angle of attack or of incidence, decreases, the center of pressure C (Fig. 81, (2)) moves
toward the front edge and tends toward a certain definite limiting position C as the angle i becomes smaller and smaller.
When a flat object like a sheet of paper is allowed to fall, it is acted upon by two forces, one W, acting at its center of gravity g, which is always vertical and equal to the weight, and the other R, which is due to the air resistance acting at the center of pressure C and perpendicular to the plane. If the plane is to fall without acceleration and without rotation, that is, if it is to glide, it is clear that these two forces must act at the same point and be equal and opposite. Hence any gliding plane must be horizontal and must move with a speed V at an angle i (see Fig. 81 (3)), given by the equation
Since the plane must be horizontal, and since there is only one angle of attack which will bring the center of pressure and the center of gravity together, it will be seen that the gliding angle i is the same for all values of the weight W, but that the speed V will be proportional to the square root of the weight (see equation 10).
FIG. 82. A stabilized glider
The foregoing theory of gliding may be nicely illustrated with paper gliders thus: Fold a sheet of writing paper lengthwise, exactly along the middle. Refold the upper half twice on itself so as to make it its original width; then fasten it down to the lower half with paste or light gummed paper. The center of gravity will now be of the new width behind the back edge of the folded portion. When started slowly with the folded edge forward, the paper will glide as described. Heavier paper will glide at the same angle but with greater speed. If started thin edge foremost, the forces at once turn the glider over, and it glides with the heavier edge in front. To increase the lateral stability it is sufficient to give the paper the shape shown in Fig. 82. (See opposite p. 317.)
airplane in flight
When the motor of an airplane stops, the plane glides safely to FIG. 83. Forces acting on an earth under the laws of equation 10. If the airplane propeller is pulling forward with a horizontal force Q, and the wings are set back at an angle i, R and W no longer balance each other, but their resultant is equal and opposite to Q; that is, the forces R, W, and form a system in equilibrium, as shown in Fig. 83. The plane moves forward horizontally with a speed V. If the angle i or the force Q is increased, the plane rises; if i or Q is diminished, the plane descends.
98. The laws of the pendulum. The first law of the pendulum was found in § 90, namely,
(1) The periods of pendulums of equal lengths swinging through short arcs are independent of the weight and material of the bobs.
Let the two pendulums of § 90 be set swinging through arcs of lengths 5 centimeters and 25 centimeters respectively. We shall thus find the second law of the pendulum, namely,
(2) The period of a pendulum swinging through a short arc is independent of the amplitude of the arc.
Let pendulums and as long as the above be swung with it. The long pendulum will be found to make only one vibration while the others are making two and three respectively. The third of the pendulum is therefore
(3) The periods of pendulums are directly proportional to the square roots of their lengths.
The accurate determination of g is never made by direct measurement, for the laws of the pendulum just established make this instrument by far the most accurate one obtainable for this determination. It is only necessary to measure the length of a long pendulum and the time t between two successive passages of the bob across the mid-point, πī and then to substitute in the formula tπ in order to obtain g with a g
high degree of precision. The deduction of this formula is not suitable for an elementary text, but the formula itself may well be used for checking the value of g, given in § 94.
QUESTIONS AND PROBLEMS
1. If a body starts from rest and travels with a constant acc `leration of 10 ft. per second each second, how fast will it be going at the close of the fifth second? What is its average velocity during the 5 sec., and how far did it go in this time?
2. A body starting from rest and moving with uniformly accelerated motion acquired a velocity of 60 ft. per second in 5 sec. Find the acceleration. What distance did it traverse during the first second? the fifth? 3. A body moving with uniformly accelerated motion traversed 6 ft. during the first second. Find the velocity at the end of the fourth second.