Let T be the tension of the string when the particle is at P; u, v the velocity when it is at A, P respectively; then

g cos 0 = resolved part of gravity in direction PO;

This gives the tension of the string in any position.

T is least when cos =1; i.e. when =0 or P is at A, and increases continually as increases, till when

=π (or

the particle is at the lowest point), T is greatest.

In order that the particle may describe a complete circle, the tension must never be negative, otherwise the string would become slack.

If we make the least value of T zero, i. e. put T=0 when 0 = 0, we get

u2

α

+g (2 − 3) = 0, or u2=ga, or u= √√(ga) ;

which expresses the least velocity the particle may have at the highest point in order to describe a complete circle.

The greatest velocity is at the lowest point, and if

u = √(ga),

the greatest velocity = √√(5ga).

The expression for the tension in this case becomes

T=3mg (1-cos ),

the maximum value of which is when cos 0-1, or when the particle is at the lowest point; the tension is then equal to 6mg = 6. weight of the particle.

The conditions necessary to be fulfilled in order that a complete circle may be described are

(i) the velocity at the lowest point must not be <√(5ga). (ii) the string must be capable of sustaining a strain equal to at least six times the weight of the particle.

110. We will conclude this chapter with a short account of the method employed by Newton to determine the elasticity of different substances.

Let A, B be two balls suspended from fixed points C, D by parallel strings, so that they may be in contact at the extremities of horizontal diameters. If the balls be drawn aside through given arcs, the velocities with which they strike each other can be found (Art. 97, Cor. 3), and by a proper arrangement of these arcs they can be made to impinge upon each other when they are in their lowest position. By observing the arcs through which they rebound, the velocities with which they separate after the impact can be obtained, and thence the coefficient of elasticity.

B

By experiments of this kind Newton determined the coefficient (e) of elasticity of balls of worsted to be about -of

5

9

balls of steel it was nearly the same,—of cork a little less,—of

lium to the Laws of Motion; where Newton further shews how allowance may be made for errors arising from the resistance of the air.

Again, if B be drawn aside and allowed to impinge upon A at rest, the velocities of each after impact will be found to be the same as result from the principles assumed in the chapter on collision.

Or again, suppose the balls to be of wood, and let one of them B have a small steel point projecting from it which would cause it to stick to A after the impact,-by properly adjusting the arcs through which the balls are displaced their velocities at impact can be made to be inversely proportional to their masses, and by loading one of them with lead their masses can be made to bear any proportion,—it will be found that they remain at rest after the impact, shewing that equal momenta in opposite directions destroy each other.

PROBLEMS AND EXAMPLES.

EXAMPLES NOT INVOLVING FRICTION. CHAPTERS I. II.

1. Two given forces act at a point; if the angle between their directions be increased, the magnitude of their resultant will be diminished, and vice versa.

2. Three given forces cannot be made to balance each other by any arrangement of their directions, if the sum of any two be less than the third.

3. Two equal forces applied at a given point have a resultant given in magnitude and direction,-find the locus of the extremity of the straight line which represents either force.

4. If O be a point within a triangle ABC, and D, E, F the middle points of the sides, the system of forces represented by OA, OB, OC will be equivalent to those represented by OD, OE, OF.

5. A circular hoop is supported in a horizontal position, and three weights P, Q, R are suspended over its circumference by three strings meeting in the centre; what must be their positions so that they may balance each other?

The angle between the directions of any two strings will be given by the formulæ of Art. 23.

6. The angles A, B, C of a triangle are 30°, 60°, 90° respectively. The point C is acted on by forces in directions

CA, CB inversely proportional to CA, CB. Find the magnitude and direction of their resultant.

Result. The resultant makes an angle 60° with CA and its magnitude : force in CA :: AB: CB.

7. If a point be acted on by three forces parallel and proportional to the three sides AB, BC, DC of a quadrilateral, shew that the resultant of the forces is represented in magnitude and direction by ECE', E being the middle point of AD, and CE' being equal to EC.

8. If two forces P and Q act at such an angle that R=P, shew that if P be doubled, the new resultant will be at right angles to Q.

9. The resultant of two forces P and Q acting on a particle is the same when their directions are inclined at an

20 as when they are inclined at an ≤

shew that tan 0 = √2 −1.

π

4

to each other:

10. A uniform sphere moveable about a fixed point in its surface, rests against an inclined plane; find the pressure on the fixed point.

Result. If a be the inclination of the plane and ẞ the angle which the radius to the fixed point makes with the vertical,

11. Two equal weights P, Q are connected by a string which passes over two smooth pegs A, B situated in a horizontal line, and supports a weight W which hangs from a smooth ring, through which the string passes. Find the position of equilibrium.

Result. The depth of the ring below the line AB