DYNAMICS.

DEFINITIONS AND PRINCIPLES.

1. A body is said to be in motion when it is continually changing its position in space.

2. Motion is said to be uniform when the spaces described in equal successive intervals of time are equal, and variable when these spaces are unequal.

3. The velocity of a body is the space it would describe in a unit of time, were the motion to become uniform at the commencement of that unit.

4. Motion is said to be accelerated when the velocity continually increases, and retarded when it continually decreases; and an accelerating or retarding force is said to be uniform or variable, according as the increments or decrements of velocity in equal times are equal or unequal.

5. The momentum or quantity of motion of a body is the sum of the motions of all its particles; and, as the motion of a particle is measured by its velocity, and the number of particles in a body constitutes its mass; hence the momentum will be equal to the product of the mass and velocity, when all the particles move with the same velocity.

6. Inertia is the opposition offered by a body to a change of state, either of rest or of motion, by the action of a force impressed upon it.

J12

7. If a system of particles, m, m1, m,... revolve round an axis, and r, r', r',....... be their respective distances from that axis; then mr2 + m ̧r12 + m2?12 + (m2) is called the moment of inertia of the system.

...or

ON THE COLLISION OF SPHERICAL BODIES.

PROP. I.

8. If a spherical body strike or act obliquely on a plane surface, the force or energy of the stroke or action, is as the sine of the angle of incidence.

Or, the force on the surface is to the same if it had acted perpendicularly, as the sine of incidence is to radius.

Let AB express the direction and the absolute quantity of the oblique force on the plane DE; or let a given body A, moving with a certain velocity, impinge on the plane at B; then its force will be to the action on the plane, as radius to the sine of the angle ABD, or as AB to BC, drawing BC perpendicular, and AC parallel to DE.

A

B

E

For, by Prop. II., the force AB is equivalent to the two forces AC, CB; of which the former AC does not act on the plane, because it is parallel to it. The plane is therefore only acted on by the direct force CB, which is to AB as the sine of the angle BAC, or ABD, to radius.

Corollary. 1. If a body act on another, in any direction, and by any kind of force, the action of that force on the second body, is made only in a direction perpendicular to the surface on which it acts.

For, the force in AB acts on DE only by the force CB, and in that direction. Corollary. 2. If the plane DE be not absolutely fixed, it will move after the stroke, in the direction perpendicular to its surface. For it is in that direction that the force is exerted.

PROP. II.

9. If one body A strike another body B, which is either at rest or moving towards the body A, or moving from it, but with a less velocity than that of A; then the momenta, or quantities of motion of the two bodies, estimated in any one direction, will be the very same after the stroke that they were before it.

For, because action and re-action are always equal, and in contrary directions, whatever momentum the one body gains one way by the stroke, the other must just lose as much in that same direction; and therefore the quantity of motion in that direction, resulting from the motions of both the bodies, remains still the same as it was before the stroke.

B

C

Thus, if A with a momentum of 10, strike B at rest, and communicate to it a momentum of 4, in the direction AB. Then A will have only a momentum of 6 in that direction; which, together with the momentum of B, viz. 4, make up still the same momentum between them as before, namely, 10.

If B were in motion before the stroke, with a momentum of 5, in the same direction, and receive from A an additional momentum of 2; then the motion of A after the stroke will be 8, and that of B, 7, which between them make 15, the same as 10 and 5, the motions before the stroke.

Lastly, if the bodies move in opposite directions, and meet one another, namely, A with a motion of 10, and B, of 5; and A communicate to B a motion of 6 in the direction AB of its motion. Then, before the stroke, the whole motion from both, in the direction of AB, is 10-5 or 5; but after the stroke, the motion of A is 4 in the direction AB, and the motion of B is 6- 5 or 1 in the same direction AB; therefore, the sum 4+ 1, or 5, is still the same motion from both as it was before.

PROP. III.

10. The motion of bodies included in a given space, is the same, with regard to each other, whether that space be at rest, or move uniformly in a right line.

For, if any force be equally impressed both on the body and the line in which it moves, this will cause no change in the motion of the body along the right line. For the same reason, the motions of all the other bodies, in their several directions, will still remain the same. Consequently, their motions among themselves will continue the same, whether the including space be at rest, or be moved uniformly forward; and therefore, their mutual actions on one another must also remain the same in both cases.

PROP. IV.

11. If a hard or fixed plane be struck by either a soft or a hard unelastic body, the body will adhere to it; but if the plane be struck by a perfectly elastic body, it will rebound from it again with the same velocity with which it struck the plane.

For, as the parts which are struck of the elastic body suddenly yield and give way by the force of the blow, and as suddenly restore themselves again with a force equal to the force which impressed them, by the definition of elastic bodies; the intensity of the action of that restoring force on the plane, will be equal to the force or momentum with which the body struck the plane. And, as action and re-action are equal and contrary, the plane will act with the same force on the body, and so cause it to rebound or move back again with the same velocity as it had before the stroke.

But hard or soft bodies, being devoid of elasticity, by the definition, having no restoring force to throw them off again, they must necessarily adhere to the plane struck.

Corollary 1. The effect of the blow of the elastic body on the plane, is double to that of the unelastic one, the velocity and mass being equal in each.

For the force of the blow from the unelastic body, is as its mass and velocity, which is only destroyed by the resistance of the plane; but in the elastic body, that force is not only destroyed and sustained by the plane, but another also equal to it is sustained by the plane, in consequence of the restoring force, and by virtue of which the body is thrown back again with an equal velocity; and, therefore, the intensity of the blow is doubled.

Corollary 2. Hence, unelastic bodies lose, by their collision, only half the motion lost by elastic bodies, their mass and velocities being equal; for the latter communicate double the motion of the former.

PROP. V.

12. If an elastic body A impinge on a firm plane DE at the point B, it will rebound from it in an angle equal to that in which it struck it; or the angle of incidence will be equal to the angle of reflection; namely, the angle ABD equal to the angle FBE.

Let AB express the force of the body A in the direction AB; which let be resolved into the two AC, CB, parallel and perpendicular to the plane. Take BE and CF equal to AC, and draw BF. D Now, action and re-action, being equal, the plane will resist the direct force CB by another BC equal

B

C

to it, and in a contrary direction; whereas, the other AC, being parallel to the plane, is not acted on nor diminished by it, but still continues as before. The body is therefore reflected from the plane by two forces BC, BE, perpendicular and parallel to the plane, and therefore moves in the diagonal BF by composition. But, because AC is equal to BE or CF, and BC is common, the two triangles BCA, BCF, are mutually similar and equal; and consequently the angles at A and F are equal, as also their equal alternate angles ABD, FBE, which are the angles of incidence and reflection.

PROP. VI.

13. Let B and b be two spherical bodies moving in the same direction with the velocities V and v; it is required to find the velocities of B and b after B has impinged on B.

b

-0

(1.) If the bodies are inelastic, it is obvious that the bodies B will move on together after impact, because there is no O elastic force to separate them. Let C be their common velocity after impact; then BV+bu is the momentum of the system, and since it must remain unchanged after impact, we must have

(2.) If the bodies B and b are perfectly elastic, the velocity lost by B and gained by b will be the same as we have found above during the compression of their figures; but after the compression ceases, elasticity begins to act, and the bodies separate with exactly the same velocity as that with which they were compressed; therefore B will lose and b will gain as much velocity by the recovery of their figures as by their compression;

hence velocity lost by B =

velocity gained by b =

b(V — v)

+

B+ b

b(V
v)
B+ b

=

Or when the bodies are perfectly elastic, we have

B+b2b:: V - v
B+b:2B:: V - v

:

velocity lost by B.
velocity gained by b.

2b(V — v)
B+ b

(3.) If the bodies are not perfectly elastic, which is usually the case, then when elasticity begins to act, it produces effects proportionally less than perfect elasticity does. Let e denote the common elasticity of the bodies; then in consequence of the restoring force, B and b will be repelled with the velocities eV and ev respectively; hence

The relative velocity of the bodies after impact is the difference of these velocities, and is hence = e(V — v).

Cor. If the bodies are moving in contrary directions before impact, attention to the signs of the velocities will preserve the truth of the formulæ above deduced, and if the body b be at rest, its velocity will be zero, and the formulæ may be readily modified to this or any other case.

EXAMPLES.

(1.) An inelastic body B impinges directly on another inelastic body C, at rest, with a velocity of 10 feet per second; find the velocity after impact, when B6 and b 4 ounces.

Here v = 0, and therefore we have, by the first case,

C=

=

BV 6.10 = 6 feet per second, the velocity after impact. B+b 6 + 4

Or, thus:

Let a common velocity after impact; then
the velocity and momentum lost by B are 10. — ≈ and 6(10 — x)
the velocity and momentum gained by C arę z and 4x.

.. 6(10 — x) = 4x, and hence x =

6, as above.

(2.) B= 10lb. is moving with a velocity 20; with what velocity must b=6lb. meet B, that their common velocity after impact may be 10 in the direction of b's motion? Ans. v 60 feet.

(3.) A sphere whose diameter is 2 inches impinges directly with a velocity of 10 feet per second on another sphere at rest, whose diameter is 4 inches; how will they move after impact?

(1) When the spheres are perfectly elastic.

(2) When their common elasticity is denoted by 18. Ans. The two-inch sphere moves backward with a velocity = 7 feet, while the other moves forward with a velocity = 23 feet. And when their elasticity is +, the velocities are respectively 73 and 24 in the same directions as in the first case.

PROP. VII.

14. If a body B impinge on b at rest, and b on b' at rest; the velocity communicated to b' will be a maximum, when b is a mean proportional between B and b'.