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THE MECHANICS OF SOLIDS
I. MOTION, VELOCITY, AND FORCE
34. Mechanics treats of the action of forces on bodies. It may be divided into two subjects, statics and dynamics. Statics treats of the laws governing forces when no motion is produced, and Dynamics or Kinetics treats of the laws governing forces by which motion is produced.
35. Motion. A body is said to have motion while it is passing continuously from one position to another. A body is at rest when its position remains unchanged.
An automobile standing by the curb is at rest. When the engine is started and its power applied, the automobile begins to move. That is, by the application of sufficient force its condition is changed from rest to motion. Rest and motion are, however, entirely relative. A body may be at rest in a railroad train, but in motion with respect to the earth. A body that is at rest with respect to the earth is in motion with respect to the sun.
The motion of a body is said to be rectilinear when it moves in a straight line. When a body moves in a path which constantly changes in direction, it is said to have a curvilinear motion, or to move in a curve. While it may not be difficult to imagine a body moving from one fixed point in space toward another without change of direction, in strict reality we know of no absolutely rectilinear motion
of bodies. A stone falling from a balloon is moving toward the center of the earth, but this is itself moving about the sun, hence the motion of the stone must be in a curve. For all practical purposes, however, a body which moves without change of direction with reference to a room or the surface of the earth is said to move in a straight line.
If a body moves over equal spaces in equal times, its motion is said to be uniform. If the distances are not equal, its motion is variable.
36. Speed; Velocity. - Speed is the rate of change of position of a moving body or its rate of motion; velocity is the speed in a definite direction. If the motion is uniform, the speed is measured by the distance the body goes in a unit of time. If the motion is variable, the speed at any instant is the distance it would move during the next unit of time if it should continue to move at the same rate.
If the speed of a body is greater for each unit of time than it was for the preceding, the motion is said to be accelerated. If the acceleration, or increase of speed, is the same for each unit of time, the motion is uniformly accelerated.
Motion is retarded, or negatively accelerated, when the speed is decreasing instead of increasing, and if the retardation is uniform, the motion is uniformly retarded.
Average or mean speed is the speed with which a body would need to move uniformly to pass over a certain space in a given time, though the actual speeds may be made up of a great many rates.
If only motion in a definite direction is considered, the above statements about speed will also hold true of velocity.
37. Space Passed Over. The space passed over by a moving body depends upon two elements, speed and time.
A train moving with an average speed of 20 mi. per hour moves 60 mi. in 3 hr. This relation may be expressed by the equation 60 = 20 X 3;
so in general,
Space passed over = Average speed X Time,
or, writing S for space passed over, v for average speed, and t for time, we have the formula
S = vt.
NOTE. The student should observe that "Space passed over = Average speed × Time" is not to be understood literally. It is merely a short and convenient way of saying, "The number of units of length passed over = the number of units of length in the average speed per unit of time X the number of units of time." The briefer wording of such formulas as this is so convenient and so commonly employed in actual use, that it will be used in this book; but the student should always bear in mind that every element, or letter, in a formula represents merely a number.
When the velocity of a body is uniformly accelerated, the rate of change in its velocity or the amount its velocity changes per second, is called its acceleration. If the body, starting from a condition of rest, has a velocity of 2 ft. per second at the end of one second, 4 ft. per second at the end of the next second, 6 at the end of the third, and so on, the gain in velocity, per second, is 2 ft. per second; that is, the acceleration is 2 ft. per second per second. In uniformly accelerated motion the average velocity for any period is half of the sum of the velocities at the beginning and end of that period. Hence in this case, as the velocity is O at the beginning, the average velocity for the first second would be 1 ft. per second, and the average velocity for the first three seconds would be 3 ft. per second.
Suppose a body to move in a certain direction from a condition of rest with a constant acceleration of a units per second per second. Its velocity per second at the end of 1 sec. will be a; at the end of 2 sec., 2 a; at the end of 3 sec., 3 a; and so on. At the end of t seconds its velocity per second will be tX a or at; that is,
Final velocity = Acceleration X Time,
v = at.
Since the velocity per second increases uniformly from O at the beginning to at at the end of t seconds, the average velocity per second for t seconds will beat, and the entire space passed over equals average velocity X time. This may be represented by the equation
at xt = at2.
1 (4) Since the velocity per second for the first second increases uniformly from 0 at the beginning to a at the end, the space passed over in that second will be a. We can get the same result by making the time 1 sec. in Equation 4, for the equation then becomes S a. We see from this that whenever a body starts from a condition of rest, and moves with a constant acceleration, the acceleration per second per second is twice the space passed over in the first second.
The space passed over during any second (the last of t seconds) may be found by subtracting from the distance passed over in t seconds the distance passed over in a time one second less. The space passed over in t seconds be
1 By combining Equations 3 and 4 we can derive the equation
S= ; and from this, v = = √2aS,
a formula sometimes convenient for finding the final velocity directly from the acceleration and the entire space passed over