17. Find the amount, the direction, and the point of application of the equilibrant of this system. (Fig. 23.)

18. Let these (Fig. 24) be considered as three weightless levers with fulcrums at F, R, and S. Weight at P = 10 lbs.

Find weight W which can be held by this combination. 19. If F and W are forces at opposite sides of a fulcrum and act in same direction at arms A and B respectively, state in a proportion the relation between F, W, A, and B to produce equilibrium.

20. Six parallel forces in same direction of 1 lb., 2 lbs., 3 lbs., 4 lbs., 5 lbs., and 6 lbs. respectively are applied to a rigid rod at points 1 ft. apart. Find their resultant. (Lock.)

21. The sum of the moments of a system of four parallel forces in equilibrium is 48. Three of the forces are as follows: Force 8, down, arm 2 from end of rod; force 16, up, arm 6 from same end; force 4, down, arm 8 from same end. Find the fourth force.

22. If x, y, and z are parallel forces at arms a, b, and c

respectively, from point O find (a) expression in terms of x, y, and z for resultant and (b) expression in terms of x, y, z, a, b, and c for arm of this resultant from 0.

XXIV. COUPLES.

1. At one end of a rod 2 meters long is a force of 5 kgm. acting due north. On the other end is a force of 5

kgm. acting due south. moments of the two forces figured about the middle point of the rod as an axis?

(a) What is the sum of the

(b) What is the sum of the moments figured from one end of the rod ?

2. Will such a system as the one in Prob. 1 tend to move N, or S, or E, or W?

3. What sort of motion will it have?

4. Suppose now to the system of Prob. 1 a north force of 10 kgm. is applied at a point 50 cm. from the original N force and 150 cm. from the original S force. Figure the three moments and find whether equilibrium results.

5. Find magnitude, point of application and direction of a fourth force which with the three of Prob. 4 will produce equilibrium.

6. To a rod extending east and west is applied a north and south couple whose arm is 80 cm. and whose forces are each 3 kgm. Find the magnitude of a couple whose arm is only 30 cm. which will produce equilibrium with the first.

7. If it is desired to produce equilibrium with the first couple whose forces are each 1 kgm., how should such a couple be applied?

8. Figure the moments of these eight forces (Fig. 25), taking O as the centre of moments.

9. Figure moments again, taking A as centre of moments.

10. Figure moments again, taking B as centre of moments.

11. Do you find the same difference between the plus and the minus moments in all three cases? How would it be if you chose any other centre ?

12. Describe a couple (i.e., give magnitudes and points

of application of its forces) which if added to the system will produce equilibrium.

13. Can more than one such couple be found?

14. State the conditions for equilibrium for both translation and rotation in systems of forces like the above.

13. Forces all in one plane, acting on a rigid body (but not all parallel). SUMMARY.-The general conditions for equilibrium in such a system are:

Resolve each force into components in any two directions at right angles to each other. Then, 1st, the algebraic sum of all the forces in each of these directions must equal zero; 2d, the sum of the moments of all the forces about any one point must equal zero. The first is the condition for no translation; the second is the condition for no rotation.

To find the resultant of such a system, find resultant in each of the directions and then the resultant of these two resultants.

XXV. FORCES IN A PLANE.

1. The force D (Fig. 26) may be resolved into a north component of BE = 3 and a west component BC= 4. Call distances above and to the right of the point A plus; below and to the left minus. Then moment of BE about A equals 3 × (2) 6. Moment of BC equals 4 x (+ 1) = 4. In the same way resolve and figure moments for each of

=

the forces. Test for translation and for rotation, and find direction and magnitude of the resultant of the four original forces.

B
A

FIG. 26.

2. Choose an axis of rotation in Fig. 27 and find resultant as in Prob. 1.

3. Find resultant of these five forces. (Fig. 28.)

FIG. 28.

B

4. Find resultant for these six forces. (Fig. 29.)

5. Find resultant for each group, and then the resultant of these two resultants. (Fig. 30.)

A

X

B

D

FIG. 30.

XXVI. CENTRE OF GRAVITY.

1. A certain uniform rod is 6 ft. long, 2 in. wide, and 1 in. Describe exactly the location of the centre of

thick. gravity.

2. A uniform rod is 18 ft. long and has a weight of 18 lbs. fastened at one end. Find the centre of gravity of the combination. Rod weighs 20 lbs.

3. At the ends of a uniform rod 50 cm. long are fastened weights of 200 and 600 gm. respectively. Where is the centre of gravity?

4. A uniform board weighing six pounds is balanced at a point 2 ft. from one end when from this end is hung a weight of 3 lbs. Find the length of the board.