272 TWO HEAVY PARTICLES CONNECTED BY A STRING.

210. We can obtain the same result by means of the equations of motion.

If T be the tension and the length of the string, these equations are

m'ï'=T¤¬x', m'ÿ =TI—Y, m'z=mg+T T2=2.

The addition of the several pairs of equations gives the first result.

Also we find as in Art. (207) that the string is always perpendicular to a certain fixed direction.

Further we have

and therefore

(x − x')2 + (y − y')2 + (z − z′ )2 = l2,

(x − x') (ï − ï') + (❀ − œ')2 + ......... = 0. (a).

and substituting in the equation (a) we deduce that T is constant.

211. Motion of a number of free particles, attracting each other with forces proportional to the distance.

In this case, by a well-known theorem, the resulting action on each particle is directed to the centre of gravity of the system, and is proportional to the distance from it. The particles therefore describe, relative to G, ellipses of which G is the centre, and in the same periodic time.

EXAMPLES.

1. Two bodies attracting each other with a force which varies inversely as the cube of the distance are projected in parallel directions; find the condition that the relative paths may be equiangular spirals.

2. If two particles of masses μ, μ attract according to the law of gravitation and be projected with velocities v, v', making an angle a with each other; shew that their orbits relative to their common centre of gravity will be parabolas, ellipses, or hyperbolas, according as

3. Two bodies, the masses of which are m and m', are projected from the points A, B, and attract each other according to the Newtonian law. The body m is projected from 4 in the direction BA with a velocity"

A

'm+m'

AB

m' is projected from B in a direction BP with a velocity

and

determine completely the path of either with regard to the other.

4. The co-ordinates (x, y), (x, y), of the simultaneous positions of two equal particles are given by the equations

prove that, if they move under their mutual attractions, the law of force will be that of the inverse fifth power of the distance.

5. Two bodies attract each other with a force varying as the distance; find the conditions that the relative orbits may be circles.

B. D.

18

6. Two particles, of M and m grammes respectively, attract according to the law of gravitation, and the relative orbit is a circle of radius a centimetres, prove that the periodic time is

If at any instant the square of the relative velocity of m be doubled, without, however, change of direction, shew that the distance apart will be doubled after an interval of time equal to

3928 √2

at
√(M +m).

seconds.

7. Two particles move under the action of their mutual attractions, one of them being constrained to remain on a fixed smooth wire in the form of a plane curve: if the path of the other be an involute to this curve and the two particles be always at corresponding points, the curve has for its intrinsic equation

s=ae

m82
2

where m is the ratio of the masses of the particles.

8. Two equal bodies attract each other with a force varying inversely as the fifth power of the distance, and they are projected with equal velocities, in opposite directions, at right angles to the line joining them; prove that there are two velocities, in the ratio of 1: √2, for each of which the relative orbits will be circles.

9. Two masses m, m' are connected by an inextensible string of length a. The extremity A to which m is attached is compelled to move with uniform acceleration in a straight line under the action of a force P in a straight line, and the extremity B to which m' is attached, is compelled to describe a circle round A with uniform angular velocity w under the action of a force Q perpendicular to AB. Find P and Q, and prove that the least value of P is

10. Two smooth circular rings (of radius a) are placed in a vertical plane with their centres in the same horizontal line at a distance 3a. Two equal beads (of mass m) slide on these rings and are connected by a thin elastic string, of which the natural length is 3a and modulus of elasticity 3 mg. They are held as far apart as possible and then let go. Find when they come to rest.

In the particular case in which λ = 1, find the whole time of the motion.

11. Two equal particles can move on a fixed smooth circular wire and attract each other with a force varying as the distance between them. Prove that their centre of gravity moves with uniform angular velocity, and that the relative motion of one with respect to the other is the same as the motion of a simple pendulum.

12. Two beads of equal mass repelling one another with a force varying inversely as the square of the distance are free to slide on a parabolic wire. If they are initially at the extremities of the latus rectum, prove that if properly projected the line joining them will always pass through the focus of the parabola.

13. The attraction between two equal particles, each of mass m, is μm2/r3, when r is the distance between them, and they are projected with equal velocities on the same side of the line (c) joining them in directions not parallel but equally inclined to that line; prove that the path of each will be an ellipse, parabola, or hyperbola, according as the initial component of each velocity in direction of the line c is less than, equal to, or greater than √2μm/c2.

14. Two small rings each of mass m, which attract each other with the force ma2 x distance, are placed on smooth wires Ox, Oy, inclined to each other at a given angle, which commence to move in their own plane with angular velocity w, and continue to move uniformly. Determine the motion of the rings.

CHAPTER XIII.

ENERGY AND MOMENTUM.

212. It is intended in this Chapter to illustrate the use of the principles of momentum and energy which were laid down in Articles (44), (47) and (52) of Chapter IV.

In many cases problems of motion are very rapidly and easily solved by the aid of these principles, and in all the cases to which they are wholly or partially applicable, the problem of determining the motion of a body or a system is reduced to the solution of equations containing simple timefluxes of the co-ordinates of the system.

213. Motion of two spheres which attract each other according to the law of nature, of given masses m and m', and given radii a and a', placed originally without kinetic energy with their centres at a given distance c from each other.

Supposing that the configuration of zero potential energy is when the spheres are in contact the potential energy of the initial configuration, which is the work done in separating the spheres,

mm'
dr

= Sata F

During the subsequent motion let u and u' be the velocities of the two balls when their centres are at a distance r from each other.