A _bevel gear_ is frequently used to change the direction of the force.
(See Fig. 94.)
[Ill.u.s.tration: FIG. 96.--A single movable pulley.]
[Ill.u.s.tration: FIG. 97.--Block and tackle.]
[Ill.u.s.tration: FIG. 98.--The fixed pulley considered as a lever.]
[Ill.u.s.tration: FIG. 99.--The movable pulley considered as a lever.]
=125. The Pulley.=--The _pulley_ consists of a wheel turning on an axis in a frame. The wheel is called a sheave and the frame a block. The rim may be smooth or grooved. The grooved rim is used to hold a cord or rope. One use of the pulley is to change the _direction_ of the acting force as in Fig. 84, where pulley _B_ changes a horizontal pull at _H_ to a downward force and pulley _A_ changes this into an upward force lifting the weight _W_. These pulleys are fixed and simply change the direction. Without considering the loss by friction, the pull at _W_ will equal that at _F_. Sometimes, a pulley is attached to the weight and is lifted with it. It is then called a _movable pulley_. In Fig. 96 the _movable pulley_ is at _P_, a fixed pulley is at _F_. When _fixed pulleys_ are used, a single cord runs through from the weight to the effort, so that if a force of 100 lbs. is applied by the effort the same force is received at the weight. But with movable pulleys several sections of cord may extend upward from the weight each with the force of the effort upon it. By this arrangement, a weight several times larger than the effort can be lifted. Fig. 97 represents what is called a _block and tackle_. If a force of 50 lbs. is exerted at _F_, each section of the rope will have the same tension and hence the six sections of the rope will support 300 lbs. weight. The _mechanical advantage of the pulley_ or the _ratio of the weight_ to the effort, therefore, _equals the number of sections of cord supporting the weight_. The fixed pulley represents a lever, see Fig. 98, where the effort and weight are equal. In the movable pulley, the fulcrum (see Fig. 99) is at _D_; the weight, _W_, is applied at the center of the pulley and the effort at _F_. The weight distance, _D_{w}_, is the radius, and the effort distance, _D_{f}_, is the diameter of the pulley. Since _W/F = D_{f} / D_{w} = 2_ in a movable pulley, the weight is twice the effort, or its mechanical advantage is 2.
Important Topics
1. Wheel and Axle, Law of Wheel and Axle.
2. Pulley, Fixed and Movable, Block and Tackle, Law of Pulley.
Exercises
1. Why do door k.n.o.bs make it easier to unlatch doors? What simple machine do they represent? Explain.
2. What combination of pulleys will enable a 160-lb. man to raise a 900-lb. piano?
3. When you pull a nail with an ordinary claw hammer, what is the effort arm? the resistance arm?
4. How much work is done by the machine in problem 2 in lifting the piano 20 ft.? How much work must be done upon the machine to do this work?
5. The pilot wheel of a boat has a diameter of 60 in.; the diameter of the axle is 6 in. If the resistance is 175 lbs., what force must be applied to the wheel?
6. Four men raise an anchor weighing {1 1/2} tons, with a capstan (see Fig. 110) having a barrel 9 in. in diameter. The circle described by the hand-spikes is {13 1/2} ft. in diameter. How much force must each man exert?
[Ill.u.s.tration: FIG. 100.--The Capstan.]
7. A bicycle has a 28-in. wheel. The rear sprocket is 3 in. in diameter,[H] the radius of the pedal crank is 7 in.; 24 lbs. applied to the pedal gives what force on the rim of the wheel? What will be the speed of the rim when the pedal makes one revolution a second?
[H] Consider the diameter of the front sprocket as 6 inches.
8. Measure the diameters of the large and small pulleys on the sewing-machine at your home. What mechanical advantage in number of revolutions does it give? Verify your computation by turning the wheel and counting the revolutions.
9. What force is required with a single fixed pulley to raise a weight of 200 lbs.? How far will the effort move in raising the weight 10 ft.?
What is the mechanical advantage?
10. In the above problem subst.i.tute a single movable pulley for the fixed pulley and answer the same questions.
11. What is the smallest number of pulleys required to lift a weight of 600 lbs. with a force of 120 lbs.? How should they be arranged?
12. A derrick in lifting a safe weighing 2 tons uses a system of pulleys employing 3 sections of rope. What is the force required?
13. Name three instances where pulleys are used to do work that otherwise would be difficult to do.
14. Draw a diagram for a set of pulleys by means of which 100 lbs. can lift 400 lbs.
(5) THE INCLINED PLANE. EFFICIENCY
=126. Efficiency.=--The general law of machines which states that the work done by a machine equals the work put into it requires a modification, when we apply the law in a practical way, for the reason that in using any machine there is developed more or less friction due to parts of the machine rubbing on each other and to the resistance of the air as the parts move through it. Hence the statement of the law that accords with actual working conditions runs somewhat as follows: _The work put into a machine equals the useful work done by the machine plus the wasted work done by it._ The _efficiency_ of a machine is the ratio of the _useful_ work done by it to the _total_ work done on the machine. If there were no friction or wasted work, the efficiency would be perfect, or, as it is usually expressed, would be 100 per cent.
Consider a system of pulleys into which are put 600 ft.-lbs. of work.
With 450 ft.-lbs. of useful work resulting, the efficiency would be 450 600 = {3/4}, or 75 per cent. In this case 25 per cent. of the work done on the machine is wasted. In a simple lever the friction is slight so that nearly 100 per cent. efficiency is often secured.
Some forms of the wheel and axle have high efficiencies as in bicycles with gear wheels. Other forms in which ropes are employed have more friction. Pulleys have sometimes efficiencies as low as 40 per cent.
when heavy ropes are used.
=127. Inclined Plane.=--We now come to a type of _simple machine of lower efficiency_ than those previously mentioned. These belong to the inclined plane group, which includes the inclined plane (see Fig. 101), the wedge and the screw. They are extensively used, however, notwithstanding their low efficiency, on account of often giving a high mechanical advantage. The _relation between these machines may be easily shown_, as the _wedge_ is obviously _a double inclined plane_. In Art.
82 it is shown that the effort required to hold a weight upon an inclined plane is to the _weight_ supported as the _height_ of the plane is to its _length_.
[Ill.u.s.tration: FIG. 101.--An inclined plane.]
Or while the weight is being lifted the vertical height _BC_, the effort has to move the length of the plane _AC_. Since by the law of machines the effort times its distance equals the weight times its distance, or the weight is to the effort as the effort distance is to the weight distance, therefore the mechanical advantage of the inclined plane is the ratio of the length to the height of the inclined plane.
_Inclined planes_ are used to raise heavy objects short distances, as barrels into a wagon, and iron safes into a building. Stairways are inclined planes with steps cut into them.
=128. The Wedge.=--Wedges are used to separate objects, as in splitting wood (see Fig. 102), cutting wood, and where great force is to be exerted for short distances. An axe is a wedge, so is a knife. A fork consists of several round wedges set in a handle. The edge of any cutting tool is either an inclined plane or a wedge. Our front teeth are wedges. Numerous examples of inclined planes may be seen about us.
No definite statement as to the mechanical advantage of the wedge can be given as the work done depends largely on friction. The force used is generally applied by blows on the thick end. In general, the longer the wedge for a given thickness the greater the mechanical advantage.
[Ill.u.s.tration: FIG. 102.--One use of the wedge.]
=129. The Screw.=--The screw is a cylinder around whose circ.u.mference winds a spiral groove. (See Fig. 103.) The raised part between the two adjacent grooves is the =thread= of the screw. The screw turns in a block called a =nut=, within which is a spiral groove and thread exactly corresponding to those of the screw. The distance between two consecutive threads measured parallel to the axis is called the =pitch= of the screw. (See Fig. 104.) If the thread winds around the cylinder ten times in the s.p.a.ce of 1 in., the screw is said to have ten threads to the inch, the pitch being {1/10} in. The screw usually is turned by a lever or wheel with the effort applied at the end of the lever, or at the circ.u.mference of the wheel. While the effort moves once about the circ.u.mference of the wheel the weight is pushed forward a distance equal to the distance between two threads (the pitch of the screw). The work done by the effort therefore equals _F 2pr_, _r_ being the radius of the wheel, and the work done on the weight equals _W s_, _s_ being the pitch of the screw. By the law of machines _F 2pr = W s_ or _W / F = (2pr) / s_. Therefore the mechanical advantage of the screw equals _(2pr) / s_. Since the distance the weight moves is small compared to that the power travels, there is a great gain in force. The screw is usually employed where _great force_ is to be exerted through small distances as in the vise (Fig. 105) the jack screw (Fig. 106), screw clamps, to accurately measure small distances as in the micrometer (Fig.
107) and spherometer, and to lessen the motion in speed-reducing devices. The worm gear (Fig. 108) is a modification of the screw that is sometimes used where a considerable amount of speed reduction is required.
[Ill.u.s.tration: FIG. 103.--The screw is a spiral inclined plane.]
[Ill.u.s.tration: FIG. 104.--The pitch is _S_.]
[Ill.u.s.tration: FIG. 105.--A vise.]
[Ill.u.s.tration: FIG. 106.--A jack screw.]
[Ill.u.s.tration: FIG. 107.--A micrometer screw.]
[Ill.u.s.tration: FIG. 108.--This large worm-wheel is a part of the hoisting mechanism employed for the lock gates of the Sault Ste. Marie Ca.n.a.l.]
Important Topics