Exercise

1. If 12 rifles cost $175, what will 40 rifles cost?

2. If a tree casts a shadow 208 feet long at the same time a post 4 feet in height casts a shadow 14 feet long, what is the height of the tree?

3. If 25 men can do a piece of work in 30 days, how long will it take 15 men to do it, working at the same rate?

4. Twenty men in 16 days can do a piece of work. How many men will it take to do it in 4 days?

5. If a train of 30 cars carries 1100 tons, how many cars must be added to the train so that the total load may be 2800 tons?

6. If a map is drawn on a scale of 100 miles to & of an inch, what distance is represented on the map by of an inch?

7. If a $32.40 tax is paid on property assessed at $2550, what tax should be paid on property assessed at $5250?

8. If 6 tons of coal cost $39, what will 10 tons cost at the same rate?

9. If in 225 pounds of milk there are 8.2 pounds of butter fat, how many pounds of milk will be required to produce 30 pounds of butter fat?

10. If 104 pounds of seed are used on a 2-acre lot, what amount will be required for 15 acres?

11. If 35 feet of sidewalk cost $40.50, what will 75 feet cost if the width is the same?

12. If a 4-foot 6-inch sidewalk costs $35, what is the cost of a 5-foot sidewalk of the same length?

13. If gun metal is composed of 2 parts of tin to 11 of copper, how many pounds of tin must be added to 190 pounds of copper for a gun-metal mixture?

14. What number bears the same ratio to 41⁄2 that 31 bears to 21?

15. What number bears the same ratio to 5.5 that 4.4 bears to 3.3?

16. If the standard 150-yard target has a bull's-eye 8 inches in diameter, what should be the diameter of a bull's-eye which will appear the same size at 25 yards?

17. If 84 pounds of metal are used in making 14 castings, how many pounds of metal will be used for 9 castings?

Such a proportion as

Direct and Inverse Proportions. 2:34:6 is called a direct proportion. In the examples in the preceding exercise the proportions are all direct.

Inverse proportion can be simply illustrated as follows:

In the figure if FP is four times the length of FW, it is evident that a pressure of ten pounds at P will counterbalance a weight of forty pounds at W. That is, 18 = 4, or

Pressure

Weight

40lb.

Length of resistance arm
Length of pressure arm

10lb.

This relation can be stated in terms of a proportion thus: The length of the pressure arm is to the length of the resistance arm as the resistance is to the pressure; or in other words, the pressure and resistance are said to be inversely proportional to the lengths of the lever arms to which they are applied. Relations involving inverse proportion are of very common occurrence.

Exercise

What load

1. The arms of a lever are 8 inches and 24 inches. on the longer arm will balance 150 pounds on the shorter arm? 2. What weight can be lifted with a lever if the pressure is 200 pounds, the pressure arm 2.5 feet, and the resistance arm 6 inches?

3. If weights of 1 pound and 12 pounds balance on a lever and the longer arm is 3 feet, what is the length of the shorter arm?

4. A lever 12 feet long balances when weights of 50 pounds and 10 pounds are hung at its ends. Find the position of the fulcrum.

5. How much does one lift on the handles of a wheelbarrow if a 100-pound load is placed in the barrow 16 inches from the

C

b

axle, and the hands 26 inches further away?

6. If an oar is 12 feet long and the rowlock is 2 feet from the handle, what force is exerted at the rowlock if the oarsman pulls

with a force of 100 pounds?

7. A safety-valve lever has its fulcrum at a, and is to push down on the valve rod at b which is one inch from a. What must be the weight of the ball c if it is to be applied at a point 12 inches from a and exert at b a force of ninety pounds?

In the mechanical world are found numerous other illustrations of inverse proportion, as for example, wherever pulleys are connected by belting, or where wheels are geared together.

-24 in.

8 in.

EXAMPLE. If a pulley 24 inches in diameter is belted to an 8-inch pulley and makes 250 revolutions per minute, how many revolutions per minute will the small pulley make?

=

Diam. of large pulley No. of revolutions per min. of small pulley Diam. of small pulley No. of revolutions per min. of large pulley Denote by x the number of revolutions per minute of the small pulley.

x=

or 24:8x: 250

24 times 250

24 × 250

8

750, the number of revolutions per minute of small pulley,

or the following rule can be used:

If

Then

R = revolutions per minute of large wheel

r = revolutions per minute of small wheel

D= diameter of large wheel

d diameter of small wheel RxD=rx d

The solution is the same as the above.

Exercise

1. If the pulley on the headstock of a lathe is 3 inches in diameter and is belted to an 8-inch pulley on a shaft that makes 420 revolutions per minute, at what rate will a block of wood placed in the lathe revolve?

2. If the pulley on the armature shaft of a dynamo is 4 inches in diameter and is belted to a driving shaft which makes 500 revolutions per minute, what must be the size of the pulley placed on the shaft if the speed of the dynamo is to be 1800 revolutions per minute?

3. A shaft has upon it two pulleys, each 8 inches in diameter. The speed of the shaft is 400 revolutions per minute. What must be the size of the pulleys of two machines if when belted to the shaft one of them has a speed of 300 revolutions per minute and the other of 900?

4. A shaft which makes 300 revolutions per minute has a 6-inch pulley upon it. This pulley is belted to the pulley on a drum 3 feet in diameter. If the drum is to wind off 1000 yards of yarn per minute, what must be the size of its pulley?

5. A cycle car has a 4-inch pulley on the driving shaft belted to a 14-inch pulley fastened to a rear wheel. If the rear wheel is 26 inches in diameter, what is the speed of the car if the engine is making 800 revolutions per minute. Allow 5 per cent for belt slippage.

Gear Ratio. The same principles as are applied to pulleys can be applied to gears. The expression gear ratio is frequently used to indicate the relative speeds of two connected wheels. The gear ratio of two connected wheels is the quotient obtained by dividing the number of revolutions the driving wheel makes by the number of revolutions the driven wheel makes in the same time.