7. Brass should be cut at 25, and wrought-iron at 22 feet per minute. Find the revolutions a lathe should turn at: Ist, when turning a brass plug 2 inches diameter; 2d, when turning a §in. wrought-iron pin.

Ans. 48 per min. nearly; 168 per min. nearly.

8. A thread is being cut on a 1-inch brass screw. Find the proper angular velocity of the work, and also the velocity at which the tool should travel to cut the thread. If the saddle be moved by a screw of 4-inch pitch, how many revolutions should it revolve at?

Ans. 10; .2 ins. per sec. very nearly; 2319 per minute. 9. The pitch of screw in a screw jack is turned by a handle 19 ins. long. of the handle to that of lifting.

10. A ship moves at 17 knots.

in miles per hour.

ins., and it is Compare the speed of the end Ans. 273: 1.

Find her speed in f.s. and
Ans. 28.7; 19 very nearly.

11. If in (10) the propeller pitch be 16 ft. Find how many revolutions the engines run at, neglecting slip. Ans. 107.3.

12. In (11) the thrust rings on the shaft are 2 ins. wide, 14 ins. external diameter. Find the maximum and mean rubbing velocities over the surface. Ans. 6.91 and 6.44 f.s.

CHAPTER II

EFFORTS AND RESISTANCES-FRICTION

THE relative motion of a pair is, in nearly all cases, resisted by some force, which we hence call the Resistance. And in order to produce the motion a force must be applied, which we call the Effort.

For example, consider the sliding pair consisting of the piston and cylinder of a steam engine.

Then the relative motion is resisted, the resistance being supplied by the connecting-rod end which bears against the end of the piston rod. The relative motion then will not take place until a sufficient effort has been applied to the piston by the steam.

We therefore now inquire into the sources from whence we derive our Efforts and Resistances in nature, and also into the way in which we are going to measure these magnitudes.

A source of effort or energy must be capable of exerting a force and of following it up. Of such sources the principal is—

Elasticity of Fluids.-By fluids we must not be understood as meaning liquids, the term fluid including liquids, gases, and vapours. It is of the latter two we speak, and they may be spoken of as elastic fluids, in distinction to liquids.

The elasticity of a fluid is the name by which we denote the power it possesses of exerting pressure on the sides of the vessel in which it is contained. If this

pressure be greater than that on the outside, then the sides, if elastic, will expand; or, more usually, the fluid is contained inside a vessel, as a cylinder, in which fits a piston, free to move, as Fig. 36.

If now the elasticity of the gas be greater than the

outside pressure, the difference of the two supplies an effort, which will move the piston out against a resistance, the effort exerting energy, and work being done on or against Atmosphere the resistance.

Fluid

If we have simply a cylinder containing a definite quantity of fluid, then this process will come to an end by gradual decrease of the elasticity till it becomes only just sufficient to balance the outside pressure. But if now Fig. 36. we apply heat to the fluid, then its elasticity will be kept up, and it will continue to exert an effort and drive the piston forward to an extent limited only by the length of the cylinder and the supply of heat. Here then we see such elasticity is one of our chief sources of effort, being the means by which the energy called Heat is utilised. For a study of the actual processes the student must consult treatises on heat engines.

The elasticity of solid bodies can also furnish us with efforts. For example, a spring wound up furnishes the effort which drives a watch.

Gravitation is our other great source of effort. Thus water falling on high ground gravitates downward, and may be used to give an effort; either by its weight being collected in an elevated reservoir, and allowed to descend on to the buckets of a water wheel; or by its motion in a stream driving the vanes of a wheel dipping in it; or where a great fall is available it may drive a turbine.

The original source of the effort is, in these cases, traceable to the heat of the sun, which originally raised the water in the form of vapour, and this is why water

is nearly always the medium whence gravitation efforts are derived. If the effort be derived from the action on a solid body we must first have lifted that body up, e.g. the weights of a clock or a pile driver. A similar remark also applies in the case of the elasticity effort when derived from a solid body, e.g. the watch spring must first have been wound up.

Again, the sun heat produces air currents, which exert efforts on the sails of windmills.

Lastly, we have the muscular efforts of living beings. In all cases the effect is-We have a force exerted driving a piece of some kind before it, producing relative motion between that piece and some other which forms a pair with it. The motion may be sliding, as in the piston and cylinder, or turning, as in the water wheel and its bearings.

Next, what are the chief resistances we meet with? The first answer to this is that the sources of effort are also sources of resistance, for taking them in order

Elasticity of a fluid or of a solid body furnishes the resistance when the work to be done is the alteration of volume of the fluid, or of shape of the solid body; e.g. the compression of air in a cylinder, the elasticity of the air resisting the sliding of the piston in the cylinder; or the elasticity of a safety-valve spring resisting the upward movement of the valve relative to its seating.

Gravitation again is perhaps the chief source of resistance, since in nearly all work the lifting of weights forms a large part, and in the raising of a weight gravitation directly resists the motion. It also is a great indirect source of resistance by causing friction.

The forces then whose sources we have considered may be equally well efforts causing, or tending to cause, relative motion; or resistances, tending to stop it. And hence they are classed as Reversible Resistances.

One kind of resistance, viz. that due to inertia, we

mention for completeness, but for the present leave out of

account.

But there is now an important class of resistance which differs essentially from the preceding, viz. Friction or Frictional Resistances.

Let us consider the motion of a sliding pair, consisting of a heavy body A on a horizontal table C, the body sliding in a guiding groove.

First, let a spring B bear against A and against a stop on the table.

But now

If now we move A towards the stop the elasticity of B supplies the resistance to the relative motion of A and C. But when we have thus compressed B its elasticity can supply an effort causing relative motion of Fig. 37. A and C. Elasticity then is reversible, and can supply either the effort causing or the resistance resisting the relative motion of the pair of bodies on each of which it acts. let there be no spring B, but suppose, which we have neglected in the preceding, that the table is rough. Then we know that if we move A towards the stop, the friction between A and C will supply a resistance to the motion. But if now, having moved A, we let it go, the friction will never move it back. And, moreover, if we now try to move A back the friction will offer just as much resistance to the return motion as it did to the first.

The friction then between two pieces of a sliding pair never tends to produce relative motion of the pair, but always to prevent it. The same holds true for all kinds of pairs, and hence we term friction an Irreversible Resistance.

Measurement of Force.-We have next to consider how to measure our efforts and resistances, i.e. forces, generally.

Force, like all other magnitudes, can only be meas