9. Find the force F, necessary to keep the weight of 200 lbs. from sliding down an inclined plane, if the dimensions and directions are as shown in Fig. 91. (Neglect friction.) Ans. 282.844+ lbs.

10. If the magnitude of the force F in Ex. 9 is 200 lbs., what is the angle between its line of action and the plane, if the force is just sufficient to prevent the weight from slipping down the plane? Ans. 45°.

=

11. Experiments indicate that a horse exerts a pull on his traces equal to about one-tenth of his weight, when the working day does not exceed 10 hours. The draft of a certain wagon is due to (a) axle friction 5 lbs. per 2000 lb. load; (b) gradient or hills; (c) rolling draft depending on height of wheel, width of tire, condition of road-bed, etc. How large a load can a team of horses each weighing 1000 lbs. pull up a 10% grade if the rolling draft is zero. (A 10% grade is a rise of 10 feet for each 100 feet along the roadway.) Ans. 1951 lbs.

100

10

FIG. 92

12. What extra pul must a horse exert on his traces (assumed horizontal) if on a level road the wheel, 4 feet in diameter, strikes a stone 2 inches high, the load being 1000 lbs. Ans. 399.4 lbs.

13. A carriage wheel whose weight is W and whose radius is r rests on a level road. Show that any horizontal force acting through the center of the wheel greater than

15. A man weighing 150 pounds stands on a ladder inclined 20° to the horizon. Find the resolved part of his weight along the ladder; perpendicular to the ladder. Ans. 51.3 lbs.; 140.85 lbs.

16. A child weighing 50 lbs. is sitting in a swing, the seat being 12 ft. from the point of support. (a) What horizontal force would be required to hold the child 6 feet to the right of the vertical through the support? (b) What would be the tension in the ropes?

Ans. (a) 28.87 lbs.; (b) 57.74 lbs.

17. Steam in the cylinder of an engine exerts a pressure of 20,000 pounds on the piston-head. The guides N, Fig. 93, are smooth. What

is the thrust in the connecting rod when it makes an angle of 20° with the horizontal? What is the pressure on the guides N?

PARALLEL FORCES ACTING IN THE SAME PLANE

18. Determine the resultant R of each of the following systems of parallel forces.

19. Let AB (Fig. 97) represent a beam carrying the weights indicated and supported by the vertical forces F1 and F2. Find F1 and F2.

20. The system of parallel forces in Fig. 98 is in equilibrium. Find the magnitudes and directions of the unknown forces F1 and F2.

1

21. A lever 12 feet long is used for raising a weight of 500 lbs. Where may the fulcrum be placed if a downward pressure of 100 lbs. is available at the other end of the lever? Ans. 2 ft. from the one end.

22. The two clevis holes of a double-tree are 16 and 20 inches from The team is pulling 270 lbs. How much does Ans. 150 lbs.; 120 lbs.

the center clevis hole. each horse pull?

23. What must be the ratio of the longer arm to the shorter arm of a double-tree (a) if one horse is to pull 10% more than the other horse; (b) if one horse is to pull 10% less than the other horse?

If the team pulls 297 lbs., how much does each horse pull in each of the above cases? Ans. (a) 10: 11; (b) 9: 10.

24. If a horse exerts a pull on his traces equal to one-tenth of his weight, where should the single-tree for each of two horses weighing 1200 and 1600 lbs., respectively, be fastened to a double-tree in order that each horse shall do his proper share of the work?

25. The end holes A and B of a double-tree are 2a inches apart. The center hole C is placed d inches in front of the mid-point of AB If

one horse is pulling ahead of the other as indicated in the figure, find what per cent. more the rear horse is pulling than the forward horse.

απ

26. The same as Ex. 25, but with a = 18 inches, d 30°.

27. With the statement of Ex. 26, is the difference between P1 and P2? each horse pull?

=

3 inches, Ans. 21.3%.

what per cent. of the total load What per cent. of the load does

1

28. Find the true weight of a body which is found to weigh 8 ounces and 9 ounces when placed in each of the scalepans of a false balance. Ans. About 8.48 ounces.

29. Find the pull P which will hold, by means of the system of pulleys shown, the given weight in equilibrium, in each of the following figures.

139. Errors.

CHAPTER VIII

SMALL ERRORS

Suppose we measure a table with a ruler, and Is it exactly that? Obviously no We can say only that we find the within a limit of accuracy beyond which we do not care or are not able to go.

find it to be 3 ft. 6 in. long. one can be certain of this. table to be 3 ft. 6 in. long

If the same table were measured using a measuring instrument whose smallest division is a hundredth of an inch we should probably find that measurements taken by two different persons would give different results, while neither result would agree exactly with our first rough measurement.

Such an inaccuracy holds for all numbers found by measurement. We cannot avoid such an approximation even when the numbers are ideal. For example, the square root of 2 is 1.4 to the nearest tenth, 1.41 to the nearest hundredth, 1.4142 to the nearest ten-thousandth; that is to say, the square of each of these numbers is nearer 2 than the square of any other with the same number of places. Similar remarks apply to all surds, to nearly all logarithms, to π, and to the various trigonometric ratios.

Whenever an attempt is made to measure the magnitude of a quantity with an observing instrument, the readings thus obtained do not represent the true value of the given magnitude. The result of a direct measurement is called an observation. The least count of the instrument is the smallest difference recognized in the readings of the instrument. Every observation is subject to an error, the error being the difference between the true value and the observed value. Errors of observation