CHAPTER VI.

MECHANICS OF SOLIDS.

XXI. REPRESENTATION OF FORCES BY LINES.

1. What three facts must be known about a force in order to describe it?

2. I wish to represent three forces, a force acting north of 6 lbs., a force south of 3 lbs., and a force northeast of 12 lbs. (a) If I represent the first by a line 4 in. long, what will be the lengths of the lines representing the other two forces? (b) What angles will the second and third lines make with the first?

3. Assuming that a force of 50 lbs. is represented by a line of 6 in., find the force represented on the same scale by a line 2 ft. long; by a line 14 in. long. (Lock.)

4. Draw two equal forces of 60 lbs. each acting at right angles to each other and having their points of application 2 ft. apart. State the scale used in representing the amounts of the forces and the scale used for the distance.

5. In Prob. 4 do the two scales have any necessary relation to each other?

6. From a straight rod AB 6 ft. long forces act up and down. The forces up are as follows: A force 6 at distance 1 ft. from A; force 8, 11⁄2 ft. from A; force 12, 1 ft. from B. The forces down are: force 18, 3 ft. from A; force 5, 2 ft. from A and a force 10 at A. Represent this system stating scales used for forces and distances.

7. Draw lines to represent six equal forces acting from a point and making equal angles with each other.

8. Draw lines to represent the forces acting on a kite. Represent the weight of the kite, the horizontal force of the wind, and the slanting force of the string. Assign values to these forces about such as you think would keep the kite at rest.

9. Draw to scale lines to represent a force of 1800 lbs. pulling a car straight along a track, a force of 200 lbs. holding it back, and a force of 100 lbs. pushing it sideways against the rails.

10. A stick weighing 3 lbs. is 8 ft. long and is supported on blocks 7 ft. apart. From the middle point is hung a weight of 10 lbs. Represent on a diagram the upward pressure of each block and the downward force of gravity at the middle of the stick.

11. Represent by a diagram all the pushes and pulls acting in the case of a picture hung on a vertical wall in the ordinary way.

12. Draw a wheel and represent two forces that would tend to turn it in the direction of the hands of a watch and three others in the opposite direction.

13. Represent the three forces acting on an oar.

14. A large wheel is turned by a small crank fastened to the axle. Represent the force turning the wheel and a force tending to hold the wheel at its rim.

15. Represent a small body suspended by six forces. Represent also the force of the weight of the body.

16. Can a single force be represented by a curved line? Give the reason for your answer.

17. A man tries to lift himself by pulling up on his boot straps. Represent the forces acting on his shoulders and on his feet. Why is there no motion?

18. Represent two forces, one twice as large as the other and acting at right angles.

19. Represent by lines the force on each rope of a child's swing and the weight of the child. Which of the three forces is the largest?

20. Name all the forces which act on a vessel moving through the water under the combined influences of steam and the wind.

COMPOSITION AND RESOLUTION OF FORCES.

Definitions.-Equilibrium; Resultant; Component; Composition; Resolution; Equilibrant.

11. Summary:

A. The resultant of two or more forces all in same straight line is their algebraic sum, calling forces in one direction +, and in opposite direction

B. The test of equilibrium of three forces acting from a point is that the lines which represent the forces must be capable of enclosing a triangle when placed tail to tip in same direction, order, and magnitude (Triangle of Forces).

c. To find resultant of two forces from a point. Construct a of which two adjacent sides shall be the two given forces. The diagonal of this from the point will then perfectly represent the resultant (Parallelogram of Forces.)

D. The test of equilibrium for any number of forces from a point is that they be capable of being represented in magnitude and direction by the sides of a polygon taken in order (Polygon of Forces.)

E. To find the resultant of three or more forces construct a figure whose sides shall represent them in order, direction, and magnitude. The line from the starting-point to the extremity of the last force so drawn will be the resultant of the system. (This line in the opposite direction is, of course, the equilibrant.) NOTE. The student will notice that statements D and E are only more general forms of B and C.

XXII. FORCES FROM ONE POINT.

1. A weight of 36 lbs. lies on a horizontal plane; a man applies an upward pull of 30 lbs. to the weight. What

is then the pressure of the weight on the plane? (Lock.) Represent by a diagram the three forces keeping the weight in equilibrium.

=

2. ABCD is a straight line. AB 2 ft.; BC= 3 ft.; CD4 ft. Find the resultant of the forces represented by AB, BC, CD, and DB. (Lock.)

3. Four spring balances are hung in a row from a single support one below the other. If each balance weighs 10 ounces, what will be the reading of each balance?

4. If now a weight of 15 ounces is hung on the lowest balance, what will each read?

5. I wish to weigh a block of 30 lbs. on spring balances.. Each of my balances will weigh 20 lbs. How can I arrange

two balances so as to weigh the 30 lbs. ?

6. A rope is stretched by a force of 100 lbs. at one end. How many pounds pull will be needed at the other end to hold it if the rope is horizontal ?

7. If the rope is vertical and weighs 50 lbs., with what force must it be held if 200 lbs. are hung on lower end? 8. If a spring balance were set into the middle of the rope in Prob. 6, how many pounds would it register?

9. How many if the rope were only half as long?

26gm,

10. In each of these cases would balances inserted in the rope at different places give different readings or the same? 11. Represent two forces of 10 kgm. acting at right angles from a single point, construct their resultant, and measure it as closely as you can.

12. Prove by a diagram. (scale in. to 1 gm.), by copying the angles as close-29gm.

ly as possible from Fig. 5,

FIG. 5.

whether these three forces produce equilibrium.

34gm.

13. Represent a force of 18 gm. acting southwest and resolve it into components south and west.

14. Represent two forces, A of 16 kgm. and B of 48 kgm., at right angles, and construct and measure resultant and equilibrant.

A

15. Four equal forces act from O as indicated (Fig. 6). LAOD is equal to twice any one of the other angles. Find by diagram the equilibrant of the system.

BY

FIG. 6.

16. AB (Fig. 7) is a track. A D horse at H pulls in the direction CH with a force of 1800 lbs. on a car at CD, find the useful component CD. 17. Copy the figure (Fig. 8), and determine whether these forces are in equilibrium.

C. If HD

=

18. ABD (Fig. 9) is an inclined plane on which rests the cask C weighing 360 lbs. CF3FG. Find the pressure against BD and the force CE needed to hold the cask.

19. Draw a triangle, and considering any two sides as forces find their resultant and their equilibrant.

20. Resolve a force of 20 gm. into two forces at right angles to each other.