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EXAMPLE. If there are two gears, A and B, running together as shown in the figure, with 32 teeth in the driving wheel A and 18 teeth in the driven wheel B, how many revolutions per minute will the driven wheel make if the driving wheel makes

B 250 revolutions per minute ?

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It is obvious that the rates of two wheels geared together are inversely proportional to the number of teeth in the wheels. That is,

Number of teeth in A Revolutions per minute of B

Number of teeth in B Revolutions per minute of A or,

No. of teeth in large wheel Revolutions per min. of small wheel No. of teeth in small wheel Revolutions per min. of large wheel

or,

No. of teeth in driving wheel
No. of teeth in driven wheel

Revolutions per min. of driven wheel
Revolutions per min. of driving wheel

Denote by r the number of revolutions per minute of the driven wheel.

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1. If a 26-tooth gear is driving a 14-tooth gear, and the 26tooth gear makes 200 revolutions per minute, at what rate is the 14-tooth gear running ?

2. A 12-inch pulley running 270 revolutions per minute drives a pulley running 90 revolutions per minute. What is the diameter of the driven pulley ?

3. A line shaft runs 250 revolutions per minute. A grindstone with a 14-inch pulley is to be belted to it. The grindstone is to run at 60 revolutions per minute. Determine the size of the pulley to be put on the line shaft so that the grindstone will run at the desired speed.

4. If the sprocket attached to the crank of a bicycle has 28 teeth and the rear sprocket over which the chain runs has 8, how many times does the rear wheel turn every time the pedals go round? How far does the bicycle advance in the same time if the diameter of the rear wheel is 28 inches ? How many times per minute must the pedals revolve when the bicycle is going at the rate of 10 miles an hour ? What is the rate of the bicycle in miles per hour when the pedals rotate 200 times per minute ?

5. In a tractor demonstration test, the following were the results. Express in each case as a per cent, the ratio of the draw bar pull to the weight of the tractor.

TRACTOR

TYPE

WEIGHT

PLOWING
SPEED

Plows

DEPTH
PLOWED

DRAWBAR

Pull

A
B
С
D
E
F
G
H
I

2-wheel
4-wheel
4-wheel
Crawler
Crawler
4-wheel
4-wheel
Crawler
3-wheel

3380
5100
3250
3500
6900
5200
7800
7400
4000

3.5
3
3
2.5
2.5
2.5
2
2.25
2.5

2–14!
3-14"
3-14"
3-14!
3-14"
4-14"
4-14"
3-14"
3-14"

9.22 8.08 8.58 8.91 9.47 8.12 7.19 8.42 8.31

2705 1510 2300 2455 3450 3125 2800 2600 2000

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CHAPTER XII

THE USE OF FORMULAS

A formula is a rule or principle expressed in algebraic symbols. Thus, the rule or principle which states that

in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides can be stated briefly in the form

C2 = a2 + 12

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where c represents the length of the hypotenuse, and a and b the lengths of the sides. In the latter form, the

rule is called a formula. 6

The rule which states that the area of a rectangle is equal to the product of the base and altitude can be expressed more briefly in a formula as follows:

A=a.6

A

where A represents area, and a and b
the sides of the rectangle.
The student has already become fa-

6 miliar in previous chapters with formulas for finding areas and volumes, such for example as A=ar, the formula for the area of a circle in terms of

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the radius; and V = arh, the formula for finding the volume of a cylinder when its radius and height are given.

The great advantage of a formula is its brevity. As an illustration of this, note the simple form in which the rule can be written which states that "the volume of the frustum of a cone or pyramid is equal to the product of one third of the height by the

sum of the lower base, the upper

FRUSTUM OF A PYRAMID base, and the square root of the product of the areas of the bases."

h The formula reads V= (a + b + Vab).

V:

3

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=

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Evaluation in a Formula. — To evaluate in a formula is to substitute for each letter its numerical value and to perform the indicated operations.

EXAMPLE. Evaluate in S= į gt?, if g = 32.2, and t = 3. Substituting in the formula the value 32.2 for g, and 3 for t, it reads :

S=} (32.2) (3)2

1 (32.2) (9)

(16.1) (9) = 144.9

=

ORAL DRILL

Evaluate in the following, using in each case the values given.

1. Arab when a = 22, b=4

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