Part 21

Mrs. B. Prolong the lines in the directions of the two forces, and you will soon discover which way the ball will be impelled; it will move from A to D, in the diagonal of a parallelogram, (fig. 7.) Forces acting in the direction of lines forming an obtuse angle, will also produce motion in the diagonal of a parallelogram. For instance, if the body set out from B, instead of A, and was impelled by the forces X and Y, it would move in the dotted diagonal B C.

We may now proceed to curvilinear motion: this is the result of two forces acting on a body; by one of which, it is projected forward in a right line; whilst by the other, it is drawn or impelled towards a fixed point. For instance, when I whirl this ball, which is fastened to my hand with a string, the ball moves in a circular direction, because it is acted on by two forces; that which I give it, which represents the force of projection, and that of the string which confines it to my hand. If, during its motion you were suddenly to cut the string, the ball would fly off in a straight line; being released from that confinement which caused it to move round a fixed point, it would be acted on by one force only; and motion produced by one force, you know, is always in a right line.

Caroline. This circular motion, is a little more difficult to comprehend than compound motion in straight lines.

Mrs. B. You have seen how the water is thrown off from a grindstone, when turned rapidly round; the particles of the stone itself have the same tendency, and would also fly off, was not their attraction of cohesion, greater than that of water. And indeed it sometimes happens, that large grindstones fly to pieces from the rapidity of their motion.

Emily. In the same way, the rim and spokes of a wheel, when in rapid motion, would be driven straight forwards in a right line, were they not confined to a fixed point, round which they are compelled to move.

Mrs. B. Very well. You must now learn to distinguish between what is called the centre of motion, and the axis of motion; the former being considered as a point, the latter as a line.

When a body, like the ball at the end of the string, revolves in a circle, the centre of the circle is called the centre of its motion, and the body is said to revolve in a plane; because a line extended from the revolving body, to the centre of motion, would describe a plane, or flat surface.

When a body revolves round itself, as a ball suspended by a string, and made to spin round, or a top spinning on the floor, whilst it remains on the same spot; this revolution is round an imaginary line passing through the body, and this line is called its axis of motion.

Caroline. The axle of a grindstone, is then the axis of its motion; but is the centre of motion always in the middle of a body?

Mrs. B. No, not always. The middle point of a body, is called its centre of magnitude, or position, that is, the centre of its mass or bulk. Bodies have also another centre, called the centre of gravity, which I shall explain to you; but at present we must confine ourselves to the axis of motion. This line you must observe remains at rest, whilst all the other parts of the body move around it; when you spin a top, the axis is stationary, whilst every other part is in motion round it.

Caroline. But a top generally has a motion forwards besides its spinning motion; and then no point within it can be at rest?

Mrs. B. What I say of the axis of motion, relates only to circular motion; that is to say, motion round a line, and not to that which a body may have at the same time in any other direction. There is one circumstance to which you must carefully attend; namely, that the further any part of a body is from the axis of motion, the greater is its velocity: as you approach that line, the velocity of the parts gradually diminish till you reach the axis of motion, which is perfectly at rest.

Caroline. But, if every part of the same body did not move with the same velocity, that part which moved quickest, must be separated from the rest of the body, and leave it behind?

Mrs. B. You perplex yourself by confounding the idea of circular motion, with that of motion in a right line; you must think only of the motion of a body round a fixed line, and you will find, that if the parts farthest from the centre had not the greatest velocity, those parts would not be able to keep up with the rest of the body, and would be left behind. Do not the extremities of the vanes of a windmill move over a much greater space, than the parts nearest the axis of motion? (plate 3. fig. 1.) The three dotted circles represent the paths in which three different parts of the vanes move, and though the circles are of different dimensions, each of them is described in the same space of time.

Caroline. Certainly they are; and I now only wonder, that we neither of us ever made the observation before: and the same effect must take place in a solid body, like the top in spinning; the most bulging part of the surface must move with the greatest rapidity.

Mrs. B. The force which draws a body towards a centre, round which it moves, is called the centripetal force; and that force, which impels a body to fly from the centre, is called the centrifugal force; when a body revolves round a centre, these two forces constantly balance each other; otherwise the revolving body would either approach the centre or recede from it, according as the one or the other prevailed.

Caroline. When I see any body moving in a circle, I shall remember, that it is acted on by two forces.

Mrs. B. Motion, either in a circle, an ellipsis, or any other curve-line, must be the result of the action of two forces; for you know, that the impulse of one single force, always produces motion in a right line.

Emily. And if any cause should destroy the centripetal force, the centrifugal force would alone impel the body, and it would, I suppose, fly off in a straight line from the centre to which it had been confined.

Mrs. B. It would not fly off in a right line from the centre; but in a right line in the direction in which it was moving, at the instant of its release; if a stone, whirled round in a sling, gets loose at the point A, (plate 3. fig. 2.) it flies off in the direction A B; this line is called a tangent, it touches the circumference of the circle, and forms a right angle with a line drawn from that point of the circumference to the centre of the circle C.

Emily. You say, that motion in a curve-line, is owing to two forces acting upon a body; but when I throw this ball in a horizontal direction, it describes a curve-line in falling; and yet it is only acted upon by the force of projection; there is no centripetal force to confine it, or produce compound motion.

Mrs. B. A ball thus thrown, is acted upon by no less than three forces; the force of projection, which you communicate to it; the resistance of the air through which it passes, which diminishes its velocity, without changing its direction; and the force of gravity, which finally brings it to the ground. The power of gravity, and the resistance of the air, being always greater than any force of projection we can give a body, the latter is gradually overcome, and the body brought to the ground; but the stronger the projectile force, the longer will these powers be in subduing it, and the further the body will go before it falls.

Caroline. A shot fired from a cannon, for instance, will go much further, than a stone projected by the hand.

Mrs. B. Bodies thus projected, you observe, describe a curve-line in their descent; can you account for that?

Caroline. No; I do not understand why it should not fall in the diagonal of a square.

Mrs. B. You must consider that the force of projection is strongest when the ball is first thrown; this force, as it proceeds, being weakened by the continued resistance of the air, the stone, therefore, begins by moving in a horizontal direction; but as the stronger powers prevail, the direction of the ball will gradually change from a horizontal, to a perpendicular line. Projection alone, would drive the ball A, to B, (fig. 3.) gravity would bring it to C; therefore, when acted on in different directions, by these two forces, it moves between, gradually inclining more and more to the force of gravity, in proportion as this accumulates; instead therefore of reaching the ground at D, as you suppose it would, it falls somewhere about E.

Caroline. It is precisely so; look Emily, as I throw this ball directly upwards, how gravity and the resistance of the air conquer projection. Now I will throw it upwards obliquely: see, the force of projection enables it, for an instant, to act in opposition to that of gravity; but it is soon brought down again.

Mrs. B. The curve-line which the ball has described, is called in geometry a parabola; but when the ball is thrown perpendicularly upwards, it will descend perpendicularly; because the force of projection, and that of gravity, are in the same line of direction.