CHAPTER V

SIMPLE MACHINES

IN order to do work we must have some source of energy, and in some instances the source can supply energy of exactly the nature required. For example, to draw a waggon we require say, a pull of 100 lbs. on the traces, this being the resistance offered by the road; then a horse is capable of exerting such a pull, and will move the waggon. In such a case

Energy exerted = work done,

[In the actual case above it is all work wasted.]

and also

Effort resistance.

But often the energy which the source can exert is of a different character to the work to be done.

example, to lift I ton through I foot,

Work done=1 ft.-ton.

For

Now a man is capable of exerting 1 ft.-ton of energy if he have time, but he cannot exert it in the form of an effort of 1 ton, but of say only 1 cwt. And then to do the above work an effort of I cwt. must be exerted through 20 ft. For the man therefore to lift I ton he must apply his effort to some part of an intermediate agent— say to the end of the rope of a set of pulleys--which agent is capable of changing the form of the energy to

that of the work, then the ton weight must be attached to some other point of the agent-in the case above to the other end of the rope-and the agent must be such as to magnify the effort 20 times, and also to reduce the velocity 20 times. Then will

I cwt. exerted through 20 ft. lift 1 ton through I ft.

We say then that the man has to use a machine, and in most cases of doing work we have to use a machine, hence the necessity of the task which we now commence, i.e. an examination of the working of various machines.

The steam engine, for example, is a machine, transmitting the energy of the steam to the point at which we require work to be done, e.g. in a colliery winding engine, to the rope which lifts the cage or coals up the shaft, and in the transmission altering the character or nature from

to

Steam pressure x distance moved by piston

Weight of coals × distance through which they are lifted.

This of course implies that there is no waste of energy in transmission, which is never the case in practice; actually some will be wasted and the remainder transmitted.

The effect which the man seeks in using a set of pulleys is the magnifying of his effort, an effect which is expressed by saying the machine gives or has a Mechanical Advantage; the magnitude of the mechanical advantage being measured by the number of times the effort is increased, i.e. by the ratio of resistance to effort.

In old treatises on machinery these latter are called generally weight and power, but we have already given a definite meaning to power, and the resistance is not in all cases a weight, so we will keep to the terms resistance, effort.

Now although the mechanical advantage is what is sought, yet there is another effect which invariably

accompanies this, i.e. the alteration of velocity. We may not desire this but we cannot avoid it, and thus the consideration of the changes of velocity, or generally of the motion of the machine, becomes a part of our subject. In some cases, e.g. a watch, motion only is the requirement. We shall commence now to examine certain machines, both as regards motion and as regards the forces to which the parts are subjected, due to the performance of work. We begin as usual with the simplest; and shall also in each case omit in the first place the influence of friction and take the case of Balanced Forces.

The Inclined Plane.-If we have a weight, say a waggon of weight W at A, which we wish to move to C, we might effect this by moving it hori

zontally to B and then lifting it direct to C.

To do this, however, we require an effort to be exerted through CB equal to the weight W, and such an effort may not be available.

W

Fig. 70.

B

We then proceed by making an inclined plane from A to C, and we shall now see that an effort much less than W will be sufficient to effect the movement.

Let now the horse pull, exerting an effort in the direction AC.

Let

P effort of horse.

Because the forces are balanced, or because we suppose the only effect produced to be the movement of the waggon (compare page 60).

This we also call the Force Ratio.

Now to examine the motion. Let

V speed of waggon along the plane.

[V may be uniform all along AC, or simply the velocity at D.]

Then V is the velocity of the effort.

Fig. 71.

C

B

Now V is compounded of a horizontal velocity V cos CAB and a vertical velocity V sin CAB.

The velocity of motion then against the resistance, or the velocity at which the waggon is rising, is V sin CAB. Velocity against resistance__V sin CAB Velocity of effort

V

= sin CAB.

This is called the Velocity Ratio, and we get the relation,

An equation which is equally true for all cases of balanced forces, and is in fact only another mode of expressing the Principle of Work. For

Energy exerted = work done

.. Px movement of P=Rx movement of R,

whence it directly follows.

The work done will be the same, whether we start from rest and end up at rest, or start with a velocity which is kept uniform during the whole motion. The only condition necessary is that we leave off at C with the same velocity with which we commenced at A. The effort, however, would, for all cases except uniform

motion, vary, and instead of P we should have P the equations.

[The student must keep the preceding statement carefully in mind, as it applies to all cases of the present chapter, and we shall therefore not specifically state it for each case.]

The results obtained have been derived by use of the Principle of Work. We will now verify them by the use of other principles which will be found explained in all treatises on Statics.

The waggon W at D being in uniform motion, the actions on it of bodies which tend to increase its motion must be exactly balanced by the actions of those bodies which tend to retard the motion.

The statement here made differs somewhat in wording from that commonly made, which is that the forces must be in equilibrium. Now it is absolutely essential that the student recognise fully from the beginning that forces can only result from the actions of other bodies on that of which we consider the motion, and we have therefore stated the principle as above.

We proceed then to question ourselves as follows:(1) What is the body acted on, and whose motion is to be considered?

Answer-The waggon.

(2) What other bodies act on it?

The answer to this in the first instance is always the same, viz. all bodies which touch it. So we put the question in the form-What other bodies touch it?

Answer The traces which pull it, and the plane. But we must always include in addition a body which acts on it without touching it, viz. the Earth.

[We do not know how the Earth acts, but we do know it does by gravitation give rise to effort and resistance. Other bodies, as magnets, also can exert actions without touching, but such actions as these are outside our present scope. ]

The full answer to (2) is then

H