2. In a combination of Wheels and Axles each of the radii of the Wheels is five times the radius of the corresponding Axle: if there be three Wheels and Axles determine what Power will balance a Weight of 375 lbs.

3. A rope, the ends of which are held by two men A and B, passes over a fixed pully L, under a moveable pully M, and over another fixed pully N. A Weight of 120 lbs. is suspended from M. Supposing the different parts of the rope to be parallel find with what force A and B must pull to support the Weight.

4. In the preceding Example if B fastens his end of the rope to the Weight find whether any change takes place in the force which A must exert.

5. A is a fixed Pully; B and C are moveable pullies. A string is put over A; one end of it passes under C and is fastened to the centre of B; the other end passes under B and is fastened to the centre of A. Compare the Weights of B and C that the system may be in equilibrium, the strings being parallel.

6. Two unequal Weights connected by a fine_string are placed on two smooth Inclined Planes which have a common height, the string passing over the intersection of the Planes; find the ratio between the Weights when there is equilibrium.

7. A Weight W is supported on an Inclined Plane by a string along the Plane. The string passes over a fixed pully, and then under a moveable pully to which a weight W is attached, and having the parts of the string on each side of it parallel; the end of the string is attached to a fixed point: shew that in order that the system may be in equilibrium the height of the plane must be half its length.

8. In an Inclined Plane if the Power P be the tension of a fine string which passes over a small fixed Pully and is attached to a Weight hanging freely, shew that if P be pulled down through a given length the height of the centre of gravity of P and W will remain unchanged.

XVIII. Virtual Velocities.

230. We have already drawn attention to a very remarkable fact with respect to machines, which is popularly expressed in these words: what is gained in power is lost in speed. This fact is included in the general Principle of Virtual Velocities, which we will now consider.

231. Suppose that A is the point of application of a force P; conceive the point A to be moved in any direction to a new position

a at a very slight distance, and from a draw a perpendicular ap on the line of action of the force P: then Ap is called the virtual velocity of the point A with respect to the force P; and the complete phrase is abbreviated, sometimes into the virtual velocity of the point A, and sometimes into the virtual velocity of the force P.

The virtual velocity is considered to be positive or negative according as p falls on the direction of P or on the opposite direction. Thus in the figure the virtual velocity is positive.

We see that Ap= Aa cos a Ap.

232. Now it is found that the following remarkable proposition is true: suppose a system of forces in equilibrium, and imagine the points of application of the forces to undergo very slight displacements, then the algebraical sum of the products of each force into its virtual velocity vanishes; and conversely if this sum vanishes for all possible displacements the system of forces is in equilibrium.

This proposition is called the Principle of Virtual Velocities.

233. We shall not attempt to demonstrate this important principle, or even to explain it fully; the student may hereafter consult the larger work on Statics. We may however notice two points.

The displacements which the principle contemplates are such as do not destroy the connexion of the points of application of the forces with each other. Thus any rigid body must be conceived to be moved as a whole, without separation into parts; also any rods or strings which transmit forces must be conceived to remain unbroken.

The word virtual is used to intimate that the displacements are not really made but only supposed to be made. The word velocities is used because we may conceive all the points of application of the forces to move into their new positions in the same time, and then the lengths of the paths described are proportional to the velocities in the ordinary meaning of this word. But there is no necessity for introducing this conception, and it would probably be advantageous for beginners if the term virtual velocity could be changed into virtual displacement.

234. In the present work we shall follow the usual course of elementary writers, and shew that the Principle of Virtual Velocities holds for all the Mechanical Powers, by special examination of each case.

Thus in every case which we shall examine there will be two forces, the Power, P, and the Weight, W; and we shall have to establish the result

Pxits virtual velocity + W × its virtual velocity=0.

We shall find that in every case the virtual velocities of P and W will have opposite signs; but as there are only two forces we shall not fall into any confusion by dropping the distinction between positive and negative virtual velocities. We shall accordingly shew that in every case we have numerically

P× its virtual velocity = W × its virtual velocity.

Suppose the

lever to be turned round C so as to come into the position aCb. Join Aa and Bb.

The angle A Ca=the angle BCb; denote it by 20.

Let CM and CN be perpendiculars from C on the lines of action of P and W respectively. Let the angle MAC=a, and the angle NBC=ß.

Then

CAa=90°-0, and CBb=90°-0;

The displacement of A resolved along AM

= Aa cos MAa= Aa cos (90°—a—0) = Aa sin (a+ 0).

=

The displacement of B resolved along NB

Therefore

= Bb cos (180o —¿BC-CBN)

= Bb cos (90°-B+0)= Bb sin (3-0).
Resolved displacement of A

Resolved displacement of B

Aa sin (a+0) CA sin (a+0)

=

Bb sin (8-0) CB sin (8-0)'

for the triangle A Ca is similar to the triangle BCb.

Now when is made indefinitely small the right hand

CA sin a

CM

side of this equation becomes

or

which is

CB sin B

CN'

W equal to

P

by the principle of the lever.

Hence ultimately, P multiplied by the resolved displacement of its point of application is equal to W multiplied by the resolved displacement of its point of applica

tion.

236.

and Axle.

To demonstrate the Principle for the Wheel

Let two circles having

the common centre C represent sections of the Wheel and Axle respectively.

Let the machine be in equilibrium with the Power P acting downwards at A, and the Weight W acting downwards at B.

Suppose the machine

to be turned round its axis so that A comes to

ь

A

a, and B to b; then aCb is a straight line.

The displacement of A resolved along the line of action of P is Ca sin A Ca; the displacement of B resolved along the line of action of W is Cb sin BCb.

Hence P multiplied by the resolved displacement of its point of application is equal to W multiplied by the resolved displacement of its point of application.