two being equal to the diameter: find the tension of the string and the pressure on the lower peg.

The tension of the string is equal to the weight and the pressure on the peg to half the weight of the sphere.

(22) One end of a uniform straight rod rests against a smooth vertical wall: a smooth ring without weight, attached to a point in the wall by a fine inextensible string, slides on the rod if be the angle which the rod, when in equilibrium,

makes with the wall, and if the length of the string be

(mm)

th

(23) A uniform heavy beam is suspended by strings attached to its extremities, passing over a smooth peg, and having unequal weights attached to their other extremities, so that the weights hang down on opposite sides of the peg: find the sides of the triangle formed by the beam and the strings, when the system is in equilibrium.

If P, Q, be the two unequal weights, W the weight of the beam and c its length, the two other sides of the triangle are equal to

(2P2 + 2 Q2 – W3) › (2P2 + 2 Q2 − W2)* *

(24) A cone, the vertical angle of which is cos, is enclosed in the circumscribing spherical surface, which is fixed; shew that it will rest in any position.

(25) A square rests with its plane perpendicular to a smooth wall, one corner being attached to a point in the wall by a string the length of which is equal to a side of the square: shew that the distances of three of its angular points from the wall are as 1, 3, and 4.

(26) A uniform isosceles triangle rests with its base horizontal on one inclined plane and its vertex on another: if a be the inclination of the latter plane and a of the former,

W. S.

12

determine the inclination of the plane of the triangle to the horizon.

If be the inclination of the plane of the triangle to the horizon,

(27) A heavy equilateral triangle, hung up on a smooth peg by a string the ends of which are attached to two of its angular points, rests with one of its sides vertical; shew that the length of the string is double the altitude of the triangle.

(28) A lamina, cut into the form of an equilateral triangle, is hung up against a smooth vertical wall by means of a string attached to the middle point of one side, so as to have a corner in contact with the wall; shew that, when there is equilibrium, the reaction of the wall and the tension of the string are independent of the length of the string, and that, if the string exceed the length of a side of the triangle, equilibrium in such a position is impossible.

CHAPTER VIII.

EQUILIBRIUM OF A SYSTEM OF BODIES.

(1) AN inextensible string binds tightly together two smooth cylinders, the ratio between radii of which is given: to find the ratio of the mutual pressure of the cylinders to the tension of the string.

Let A, B, fig. (83), be the centres of transverse circular sections of the cylinders, in the plane of the string: produce the common tangent HK to meet AB, produced, in C: from A, B, draw AH, BK, to the points of contact H, K: draw KL, parallel to BA, to meet AH in L. Let R denote the action of the cylinder A upon the cylinder B, T being the tension exerted upon the latter cylinder by each of the two rectilinear portions of the string. Let r, s, be the radii of A, B, respectively, and let ACH 0. Then, for the equilibrium of the cylinder B, we have, resolving along AC,

=

(2) Three equal heavy cylinders, each of which touches the other two, are bound together by a string and laid upon a horizontal plane; the tension of the string being given, to find the pressures between the cylinders.

Let W be the weight of each cylinder, T the tension of the string, R the reaction between the higher and each of the lower cylinders, S the reaction between the two lower, R' the reaction of the horizontal plane upon each of the lower cylinders. The forces acting upon each cylinder are exhibited in diagram (84).

The two forces T, T, which act upon the higher cylinder, may be replaced by a vertical force 2 T cos 30°, acting downwards in the line of W's action: hence, for the equilibrium of the higher cylinder, we have, resolving forces vertically,

2R cos 30° = 2T cos 30° + W,

or

R=T+

W √3

(1).

Again, the two forces T, T, which act upon either of the lower cylinders, may be replaced by a force 27'cos 30o, the direction of which bisects the angle between R and S: hence, resolving horizontally the forces which act on one of the lower cylinders, we have

The equations (1) and (3) give the pressures between the cylinders.

COR. Resolving vertically the forces which act on one of the lower cylinders, we have

R'+2T cos 30°. sin 30° W + R sin 60°,

=

Thus the sum of the pressures on the horizontal plane is equal to 3 W, the weight of the three cylinders; as might have been anticipated.

(3) A uniform beam is moveable round a hinge, which is fixed on a smooth inclined plane: the other end presses against a right-angled cone, of equal weight, resting on the plane, with which its base is in contact: the inclination of the beam to the plane being the same as that of the plane to the horizon, to find the inclination of the plane to the horizon.

Let R, fig. (85), be the mutual action between the beam and the cone: let W be the weight of the cone or beam: and ✪ the inclination of the plane to the horizon.

Then, for the equilibrium of the beam, taking moments about the hinge, we have, 2a being the length of the beam,

and, for the equilibrium of the cone, resolving the forces, which act on it, parallel to the plane,

and therefore the plane's inclination to the horizon is equal to tan‍1 (3).

(4) Two equal uniform rods AA', BB', fig. (86), are attached to smooth hinges at A, B, in a horizontal line, their lower ends being tied together by a fine string: a sphere is placed upon the two rods to find the tension of the string.

Let R be the mutual action between each rod and the sphere at the point of contact E. Let W be the weight of the sphere and P of each rod. Let T be the tension of the string.