EXAMPLES.

A power of 1 lb., acting parallel to an inclined plane, supports a weight of 2 lbs. What is the inclination of the plane? Ans. 30°.

2. The power, resistance, and normal pressure, in the case of an inclined plane, are, respectively, 9, 13, and 6 lbs. What is the inclination of the plane, and what angle does the power make with the plane?

SOLUTION.

If we denote the angle between the power and resistance by, and the inclination of the plane by a, we shall have, from Art. (35),

6 = √/132 + 92 + 2 × 9 × 13 cos ;

156° 8' 20".

Also, from Art. (35), for the inclination of the plane,

3. A body may be supported on an inclined plane by a force of 10 lbs., acting parallel to the plane; but it requires a force of 12 lbs. to support it when the force acts parallel to the base. What is the weight of the body, and what is the inclination of the plane?

Ans. The weight is 18.09 lbs., and the inclination is 33° 33' 25".

The Pulley.

86. A pulley consists of a wheel having a groove around its circumference to receive a cord; the wheel turns freely on an axis at right angles to its plane, which axis is supported by a frame called a block. The pulley is said to be fixed, when the block is fixed, and to be movable, when

the block is movable. Pulleys may be used singly, or in combinations.

87.

Single fixed Pulley.

T

In this pulley the block, and, consequently, the axis, is fixed. Denote the power by P, the resistance by R, and the radius of the pulley by r. It is plain that both the power and resistance should be in a plane, at right angles to the axis. Hence, if we take the axis of the pulley as the axis of moments, we shall have (Art. 49), the following condition of equilibrium:

Pr= Rr; or, P = R.

AB

P

Ꭱ

Fig. 75.

That is, in the single fixed pulley, the power is equal to the resistance.

The effect of the pulley is, therefore, simply to change the direction of the force, and it is for this purpose that it is generally used.

Single Movable Pulley.

88. In this pulley the block, and, consequently, the axis, is movable. The resistance is applied at a hook attached to the block; one end of a rope, enveloping the lower part of the pulley, is firmly attached at a fixed point C, and the power is applied at the other extremity. We shall take the two branches of the rope parallel, that being the most advantageous way of using the machine.

Adopting the notation of the preceding article, and taking A, the point of contact of CA with the pulley, as the centre of moments, we shall have, for the condition of equilibrium (Art. 49),

P × 2r=

Rr;

..

P = R.

R

Fig. 76.

GE B

That is, in the movable pulley, when the power and resistance are parallel, the power is equal to one half of the resistance. The tension upon the cord CA is evidently the

same as that upon the cord BP. It is, therefore, equal to the power, or to one-half the resistance. If, therefore, the resistance of the fixed point C be replaced by a force equal to P, the equilibrium will be undisturbed.

If the two branches of the enveloping cord are oblique to each other, the condition of equilibrium will be somewhat modified. Suppose the resistance of the fixed point C to be replaced by a force⚫ equal to P, and denote the angle between the two branches of the

cord by 2p. If an equilibrium subsists between the forces P, P, and R, we must have the relation,

2 Pcosp = R.

R

Fig. 77.

Draw the cord AB between the points of contact of the cord and pulley, and denote its length by c; draw, also, the radius OB. Then, since OR is perpendicular to AB and BP to OB, the angle ABO will be equal to one half of the angle ABC, or equal to . Hence,

That is, the power is to the resistance as the radius of the pulley is to the chord of the arc enveloped by the rope.

When the chord is greater than the radius, there will be a gain of mechanical advantage in the use of this pulley; when less, there will be a loss of mechanical advantage.

If the chord becomes equal to the diameter, we have, as before,

P = R.

89.

Combinations of Separate Movable Pulleys.

The figure represents a combination of three movable pulleys, in which there are as many separate cords as there are pulleys; the first end of each cord is attached at a fixed point, the second end being fastened to the hook of the next pulley in order, except the last cord, at the second extremity of which the power is applied.

Let us denote the tension of the cord between the first and second pulley by t, that of the cord between the second and third pulley by t'. By the preceding Article, we have,

VR

Fig. 78.

Multiplying these equations together, member by member, and striking out the common factors in the resulting equation, we have,

Had there been n pulleys in the combination, we should have obtained, in an entirely similar manner, the relation,

P = (1)". R; .. P: R: : 1 : 2n (42.)

That is, the power is to the resistance as 1 is to 2o, n denoting the number of pulleys.

For convenience, the last branch of the cord is often passed over a fixed pulley; this arrangement only serves to change the direction of the force, without in any way changing the conditions of equilibrium.

Combinations of Pulleys in blocks.

90. These combinations are effected in a variety of ways. In most cases, there is but a single rope employed, which, being firmly attached to a hook of one block, passes around a pulley in the other block, then around one in the

first block, and so on, passing from block to block until it has passed around each pulley in the system. The power is applied at the free end of the rope. Sometimes the pulleys in the same block are placed side by side, sometimes they are placed one above another, as represented in the figure, in which case the interior pullies are made somewhat smaller than the outer ones. The conditions of equilibrium are the same in both cases. To deduce the conditions of equilibrium in the case represented, in which the upper block is fixed and the lower one movable denote the power by P, the resistance by R. When there is an equilibrium between P and R, the tension upon each branch of the rope which aids in supporting the resistance must be the same, and equal to P; but, since the last pulley simply serves to change the direction of the force P, there will be four such branches in the case considered; hence, we shall have,

P

Ꭱ

Fig. 79.

Had there been n pulleys in the combination, there would have been n supporting branches of the cord. and we should have had, in the same manner,

That is, the power is to the resistance as 1 is to the number of branches of the rope which support the resistance.

The principles involved in the combinations already considered, will be sufficient to make known the relation between the power and resistance in any combination what

ever.

EXAMPLES.

1. In a system of six movable pulleys, of the kind described in Art. 89, what weight can be sustained by a power of 12 lbs? Ans. 768 lbs.