of support, it may be seen that the condition of equilibrium is that the moment Aa shall be equal to the moment Bb.

See Fig. 2, in which vertical forces A and B have the same direction and act upon the horizontal lever at points situated at distances a and b on opposite sides of the fulcrum F. In Fig. 3 the forces A and B have opposite directions and act on the horizontal lever at points which are situated on the same side of the fulcrum F.

26. A horizontal bar in equilibrium will remain at rest if vertical forces, represented by A, B, C, D, E, etc., acting at distances a, b, c, d, e, etc., from the fulcrum F, satisfy a conditional equation such as Aa + Cc Bb + Dd + Ee.

27. It is a principle that a mass may be treated in calculations as if it were concentrated at a certain point called the center of gravity of the mass.

EXERCISE XVI. 8

Solve each of the following problems:

1. How heavy a stone can a man, by exerting a force of 160 pounds, lift with a crowbar 6 feet in length, if the fulcrum be one foot from the stone (neglecting the weight of the crowbar)?

2. A wheelbarrow is loaded with 50 bricks, each weighing 6 pounds. What lifting force must be applied at the handles to raise the load (neglecting the weight of the wheelbarrow), provided that the center of gravity of the load is 2 feet from the center of the wheel and the hands are placed at a distance of 4 feet from the center of the wheel?

3. A beam 20 feet in length and weighing 50 pounds is supported at a point 4 feet from one end. What force must be applied at the end farthest from the point of support to keep the beam in equilibrium? What force must be applied at the end nearer the point of support?

4. A board 15 feet in length and weighing 21 pounds is supported at a point 2 feet from the center. If the board is kept in equilibrium by a stone placed on it at a point 3 feet from the fulcrum, find the weight of the

stone.

5. A horizontal bar 18 inches in length is in equilibrium when forces of 4 pounds and 2 pounds respectively are acting downward at its ends. Find the position of the point of support.

6. A basket weighing 100 pounds is suspended at a point two feet from the end of a stick which is 8 feet in length and which weighs three pounds. If the stick is being carried by two boys, one at each end, how many pounds does each boy lift?

7. Two boys, one at each end of a stick 12 feet in length which weighs 5 pounds, raise a certain weight which is suspended from the stick. How heavy is the weight and at what point does it hang, if one boy lifts 35 pounds and the other lifts 30 pounds?

Since the boys lift 35 and 30 pounds respectively, and the weight of the stick is 5 pounds, it follows that the weight carried must be 60 pounds.

If the stick be assumed to be uniform, it may be seen that one boy will carry 32 pounds and the other boy 27 pounds of the weight.

We will represent by z the number of feet from the center of gravity of the stick to the point at which the weight is suspended on the side of the center of gravity nearer the boy exerting the greater force.

It may be seen that, with respect to the center of gravity, regarded as a fixed point, the weight of 60 pounds which is carried and the force of 27 pounds exerted at one end of the stick both tend to produce rotation of the stick about its center of gravity in one direction, while the force of 32 pounds exerted at the other end of the stick tends to produce rotation of the stick about its center of gravity in the opposite direction.

Since the stick is in equilibrium, the sums of the moments of these forces must be equal.

Hence we have the following conditional equation:

60x + (27) × 6 = (321) × 6.

Solving, we obtain x =

Hence, the weight is suspended from the stick at a point which is 6 inches from the center of gravity on the side nearer the boy exerting the greater force.

This value will be found to satisfy the conditions of the given problem. 8. A safety valve having an area of 4 square inches is held down by a lever which is hinged at one end.

The lever is 10 inches long and the point of application of the valve is 2 inches from the hinged end of the lever. If a weight of 12 pounds is placed on the free end, find the pressure per square inch on the valve which will lift the safety valve, disregarding the weight of the lever.

9. A dog-cart carrying a load of 576 pounds is found, when on a level road, to exert a pressure of only 8 pounds on the horse's back. If the dis

of support, it may be seen that the condition of equili the moment Aa shall be equal to the moment Bb.

See Fig. 2, in which vertical forces A and B ha direction and act upon the horizontal lever at poin distances a and b on opposite sides of the fulcrun the forces A and B have opposite directions and a zontal lever at points which are situated on the fulcrum F.

a

a

FIG. 2.

FI

26. A horizontal bar in equilibrium will rema forces, represented by A, B, C, D, E, etc., acti c, d, e, etc., from the fulcrum F, satisfy a condi as Aa + Cc = Bb + Dd + Ee.

27. It is a principle that a mass may be t as if it were concentrated at a certain point gravity of the mass.

EXERCISE XVI. 8

Solve each of the following problems:

1. How heavy a stone can a man, by exerti lift with a crowbar 6 feet in length, if the ful stone (neglecting the weight of the crowbar)?

2. A wheelbarrow is loaded with 50 bricks What lifting force must be applied at the hand ing the weight of the wheelbarrow), provide of the load is 2 feet from the center of the wh at a distance of 4 feet from the center of the w

3. A beam 20 feet in length and weighing point 4 feet from one end. What force must from the point of support to keep the beam must be applied at the end nearer the point

4. A board 15 feet in length and weigh a point 2 feet from the center. If the bo stone placed on it at a point 3 feet from the

stone.

b

CHAPTER XVII

SIMULTANEOUS LINEAR EQUATIONS

GENERAL PRINCIPLES OF EQUIVALENCE

1. Two or more conditional equations are said to be simultaneous with reference to two or more unknowns appearing in them when each unknown letter is assumed to represent the same number wherever it appears in all of the equations.

2. A set or group of simultaneous equations is called a system of simultaneous equations.

E. g. The equations 3x + 2y = 14 (1) and x+5y = 9 (2) are simultaneous on condition that x represents the same number in (1) as in (2), and that y has the same value in one equation as it has in the other.

3. A solution of a conditional equation containing two or more unknowns is any set of values which, when substituted for the unknowns, reduces the conditional equation to an identity.

E. g. The sets of two values, x = 2, y = 4; x = : 0, y = 7; x = 6, y = − 2, etc.; are solutions of the conditional equation 3 x + 2 y = 14 containing two unknowns.

4. A solution of a system of simultaneous conditional equations is any set of values of the unknowns which satisfies all of the equations of the system.

E. g. The single set of two values x = 4, y = 1 is the single solution of the system of two simultaneous conditional equations 3x + 2y = 14 (1), and x+5y=9 (2).

5. The word "solution" may be used to denote either the process of solving an equation or system of equations, or the value or values obtained by the process.

6. If the number of solutions, that is, the number of different sets of values which satisfy all of the equations of a given system. of simultaneous conditional equations, — is limited or finite, the system is said to be determinate.

If, however, the number of different sets of values which satisfy all of the equations of a system be unlimited or infinite, the system is said to be indeterminate.

7. Two conditions restricting the values of two or more unknown numbers are said to be consistent if both conditions can be satisfied by the same values of the unknowns.

In the contrary case, the conditions are said to be inconsistent.

E. g. If we are required to find two numbers whose sum is 10 and difference 8, the conditions restricting the values of the unknown numbers are consistent, since we can find two numbers, 9 and 1, which satisfy them.

Two conditions requiring that the sum of two unknown numbers shall be 10 and also 8 are inconsistent, since it is impossible to find two such numbers.

8. The graph of a conditional equation of the first degree containing two unknowns is a straight line.

This may be shown directly by applying certain simple properties of plane triangles.

(The following proof is offered for such students as are acquainted with a few of the simple principles of geometry, and may be omitted when the chapter is read for the first time.)

Let A and A' represent any two points on the graph of a given equation y = ax, located by means of the coördinates (x, y) and (x', y'), so taken that corresponding values of x and y satisfy the given equation. (See Fig. 1.)

Draw straight lines from the origin 0 to A and A'.

Since the values x, y, and x', y', are assumed to satisfy the equation y = ax, we have y = ax and y' = ax'.

The ordinates y and y' are taken parallel to the axis of Y, and accordingly the triangles OBA and OB'A' are similar, since they have an angle of one equal to an angle of the other, and the included sides proportional. The corresponding angles AOB and A'OB' are consequently equal, and the lines OA and QA' coincide.