1. If a substance be weighed in a balance having unequal arms, and in one scale appear to weigh a lbs. and in the other scale b lbs., shew that its true weight is √(ab) lbs.

2. A body, the weight of which is one lb., when placed in one scale of a false balance appears to weigh 14 ounces: find its weight when placed in the other.

3. The arms of a balance are in the ratio of 19 to 20; the pan in which the weights are placed is suspended from the longer arm: find the real weight of a body which apparently weighs 38 lbs.

4. If a balance be false, having its arms in the ratio of 15 to 16, find how much per lb. a customer really pays for tea which is sold to him from the longer arm at 3s. 9d. per lb.

The following six questions relate to the common steelyard:

5. If the moveable weight for which the steelyard is constructed be one lb., and a tradesman substitutes a weight of two lbs., using the same graduations, shew that he defrauds his customers if the centre of gravity of the steelyard be in the longer arm, and himself if it be in the shorter arm.

6. The moveable weight is one lb., and the weight of the beam is one lb.; the distance of the point of suspension from the body weighed is 2 inches, and the distance of the centre of gravity of the beam from the body weighed is 3 inches: find where the moveable weight must be placed when a body of 3 lbs is weighed.

7. If the fulcrum divide the beam, supposed uniform, in the ratio of 3 to 1, and the weight of the beam be equal to the moveable weight, shew that the greatest weight which can be weighed is four times the moveable weight.

8. If the beam be uniform and its weight of the

m

moveable weight, and the fulcrum be of the length of the

т. м.

n

8

beam from one end, shew that the greatest weight which can 2m (n-1)+n-2

be weighed is

2m

times the moveable weight.

9. Find what effect is produced on the graduations by increasing the moveable weight.

10. Find what effect is produced on the graduations by increasing the density of the material of the beam.

11. A straight uniform lever whose weight is 50 lbs. and length 6 feet, rests in equilibrium on a fulcrum when a weight of 10 lbs. is suspended from one extremity: find the position of the fulcrum and the pressure on it.

12. Two weights P and Q hang at the ends of a straight heavy lever whose fulcrum is at the middle point; if the arms are both uniform, but not of the same weight, and the system be in equilibrium, shew that the difference between the weights of the arms equals half the difference between P and Q.

13. A uniform heavy rod AB, seven feet long, is supported in a horizontal position between two pegs C and D, two feet apart, of which C is half a foot from the end 4: find the pressures on the pegs.

If a force act upwards at a distance of half a foot from the end B, sufficient to remove all pressure from the peg C, shew that the pressure on the peg D will be half of what it was before.

14. Two men carry a uniform plank 6 feet in length and weighing 2 cwt., and at 2 feet from one end a weight of 1 cwt. is placed; if one man have this end resting on his shoulder, find where the other man must support the beam in order that they may share the whole weight equally.

15. Two weights of 2 lbs. and 5 lbs. balance on a uniform heavy lever, the arms being in the ratio of 2 to 1: find the weight of the lever.

16. If a heavy uniform rod c inches long be supported on two props at distances a and b inches from the ends, compare the pressures on the props.

17. A uniform heavy bar ten feet long and of given weight W is laid over two props in the same horizontal line, so that one foot of its length projects over one of the props. Find the distance between the props so that the pressure on one may be double that on the other. Also find the pressures.

18. A straight lever weighing 20 lbs. is moveable about a fulcrum at a distance from one extremity equal to onefourth of its length: find what weight must be suspended from that extremity in order that the lever may remain at rest in all positions.

19. A bent lever is composed of two straight uniform rods of the same length, inclined to each other at 120°, and the fulcrum is at the point of intersection: if the weight of one rod be double that of the other, shew that the lever will remain at rest with the lighter arm horizontal.

20. A bar of iron of uniform section and 12 feet long is supported by two men, one of whom is placed at one end: find where the other must be placed so that he may sustain three-fifths of the whole weight.

21. A cylindrical bar of lead a foot in length, and 8 lbs. in weight is joined in the same straight line with a similar bar of iron 15 inches long and 6 lbs. in weight: find the point on which they will balance horizontally.

22. A uniform rod 10 feet long and 48 lbs. in weight is supported by a prop at one end: find the force which must act vertically upwards at a distance of 2 feet from the other end to keep the rod horizontal.

23. A straight uniform rod is suspended by one end: determine the position in which it will rest when acted on by a given horizontal force at the other end.

24. Two weights acting perpendicularly on a straight uniform lever at its ends on opposite sides of the fulcrum balance: if one weight be double the other, but the weight of the lever equal to their sum, find where the fulcrum must be.

XIII. The Wheel and Axle. The Toothed Wheel.

180. The present Chapter will be devoted to the Wheel and Axle, and the Toothed Wheel. It will be seen that these two Mechanical Powers are only modifications of the Lever.

W

181. The Wheel and Axle. This machine consists of two cylinders which have a common axis; the larger cylinder is called the Wheel, and the smaller the Axle. The two cylinders are rigidly connected with the common axis, which is supported in a horizontal position so that the machine can turn round it. The Weight acts by a string which is fastened to the axle and coiled round it; the Power acts by a string which is fastened to the wheel and coiled round it. The Weight and the Power tend to turn the machine round the axis in opposite directions.

182. When there is equilibrium on the Wheel and Axle, the Power is to the Weight as the radius of the Axle is to the radius of the Wheel.

Let two circles having the common centre C represent sections of the wheel and axle respectively, made by planes perpendicular to the axis of the cylinder.

It may be assumed, that the effects of the Power and the Weight will not be altered if we suppose them both to act in the same plane perpendicular to the axis. Let the string by which the power, P, acts leave the wheel at A, and the string by which the weight, W, acts leave the axle at B. Then CA and CB will be perpendicular to the line of action of P and W. We may regard

ACB as a lever of which C is the fulcrum, and hence, by Art. 165, the necessary and sufficient condition for equilibrium is

183. If we wish to take into account the thickness of the strings by which P and W act, we may consider that the line of action of each of these forces coincides with the middle of the respective strings. Thus, in the condition of equilibrium, CA will denote the radius of the wheel increased by half the thickness of the string by which P acts, and CB will denote the radius of the axle increased by half the thickness of the string by which Wacts.

184. We have supposed that the Power in the Wheel and Axle acts by means of a string; but the Power may act by means of the hand, as in the familiar example of the machine used to draw up a bucket of water from a well.

A windlass and a capstan may also be considered as cases of the Wheel and Axle.

The windlass scarcely differs from the machine used to draw up water from a well: the windlass however has more than one fixed handle for the convenience of working