resultant is 20 lbs. What are the intensities of the compo nents? SOLUTION. X= We have, 3X=4Y, or X = Y, and R = 20; Hence, 2 2 20 = √X2 + 2X2 = {X; X = 12, and Y = 16. 3. A boat fastened by a rope to a point on the shore, is urged by the wind perpendicular to the current, with a force of 18 pounds, and down the current by a force of 22 pounds. What is the tension, or strain, upon the rope, and what angle does it make with the current? Hence the tension is 28.425 lbs., and the angle 39° 17' 28". IZ N Components of a Force in the direction of three axes. 31. To find expressions for the components of a force in the directions of three rectangular axes. Let OR represent the force, and OX, OY, and OZ, three rectangular axes drawn through its point of application, 0. Construct a parallelopipedon on OR as a diagonal, having three of its edges coinciding with the axes. Then will the lines OL, OM, and ON, represent 0 ΔΙ Fig. 11. the required components. Denote these components, respectively, by X, Y, and Z. Draw lines from R, to L, M, and N, respectively; these will be perpendicular to the axes, and with them, and the force R, will form three right-angled triangles. Denote the angle between R and the axis of X by a, that between R and the axis of Yby ß, and that between R and the axis of Z by 7; we shall have from the right-angled triangles referred to, the following equations: 0 Fig. 11. X = R cos a, Y = R cos B, and Z = R cos y. The angles a, B, and y, are estimated from the directions of the positive co-ordinates, through 360°. The components above found will be positive when they act in the direction of positive co-ordinates, and negative when they act in a contrary direction. If we regard X, Y, and Z, as three forces, R will be their resultant, and we shall have, from a known property of the rectangular parallelopipedon, That is, the resultant of three forces at right angles to each other, is equal to the square root of the sum of the squares of the components. We also have from the figure, Hence, the position of the resultant is completely determined. EXAMPLES. 1. Required the intensity and direction of the resultant of three forces at right angles to each other, having the intensities 4, 5, and 6 pounds, respectively. whence, a = 62°53', B = 55°15'32", and y = 46°51'31". Hence, the resultant pressure is 8.775 lbs., and it makes, with the components taken in order, angles equal to 62° 53', 55° 15' 32", and 46° 51′ 31′′. 2. Three forces at right angles are to each other as the numbers 2, 3, and 4,,and their resultant is 60 lbs. What are the intensities of the forces? SOLUTION. We have Hence, 60 = Y = 3X, Z = 2X, and R = 60; 'X2 + 2X2 + 4X2 = X√29 = 2.6925 X .. X = 22.284. The components are, therefore, 22.284 lbs., 33.426 lbs., and 44.568 lbs. Projection of Forces. 32. If planes be passed through the extremities of a force, perpendicular to the direction of any straight line, that portion of the line intercepted between them is the projection of the force upon the line. The operation of resolving forces into components in the direction of rectangular axes, is nothing more than that of finding their projections upon these axes. If two straight lines be drawn through the extremities of a force, perpendicular to any plane, and the points in which they meet the plane be joined by a straight line, this line is the projection of the force upon the plane. If we denote any force by P, and the angle which it makes with any line or plane by a, P cos a will represent the projection of the force on the line or plane. In both cases the projection of the force is its effective component in the direction of the line or plane upon which it is projected. Compositon of a Group of Forces in a Plane. 33. Let P, P', P", &c., denote any number of forces lying in the same plane, and applied at a common point, and represent the angles which they make with the axis of X by a, a', a", &c. Their components in the direction of the axis. of X are P cos a, P' cos a', P" cos a", &c., and their components in the direction of the axis of Y, are P sin a, P' sin a', P" sin a", &c. If we denote the resultant of the group of components which are parallel to the axis of X by X, and the resultant of the group parallel to the axis of Y by Y, we shall have, (Art. 26), Σ (P sin a).. (5.) X= (P cos a), and Y The resultant of X and Y is the same as the resultant of the given forces. Denoting this resultant by R, and recollecting that X and Y are perpendicular to each other, we have, as in Article 30, If we denote the angle which the resultant makes with the axis of X by a, we shall have, as in Article 30, 1. Three forces, whose intensities are respectively equal to 50, 40, and 70, lie in the same plane, and are applied at the same point, and make with an axis through that point, angles equal to 15°, 30°, and 45°, respectively. Required the intensity and direction of the resultant. SOLUTION. We have, X = 50 cos 15° + 40 cos 30° + 70 cos 45° = 132.435, and Y = 50 sin 15° + 40 sin 30° + 17 sin 45° = 82.45; whence, The resultant is 156, and the angle which it makes with the axis is equal to 31° 54' 24". 2. Three forces 4, 5, and 6, lie in the same plane, making equal angles with each other. Required the intensity of their resultant and the angle which it makes with the least force. SOLUTION. Take the least force as the axis of X. Then the angle between it and the second force is 120°, and that between it and the third force is 240°. We have X = 4 + 5 cos 120° + 6 cos 240° — - 1.5; .866; 3. Two forces, one of 5 lbs. and the other of 7 lbs., are applied at the same point, and make with each other an angle of 120°. What is the intensity of their resultant? Ans. 6.24 lbs. 34. Composition of a Group of Forces in Space. Let the forces be represented by P, P', P", &c. The angles which they make with the axis of X, by a, a', a', &c., the angles which they make with the axis of Y, by ß, B', B", &c., and the angles which they make with the axis |