Also, since n: 1 :: area ABC: area Abc :: AD3 : Aď3,

1848.

3√n. (√n + 1)

(A). When a weight is supported on a smooth inclined plane by a force along the plane, the force is to the weight as the height of the plane is to its length.

(B). If the roughness of a plane, which is inclined to the horizon at a known angle, be such that a body will just rest supported on it, find the least force along the plane requisite to drag the body up.

In this case we have an extension of the problem (4) to the case where the plane is rough, the roughness being given by the fact that a body would just rest on the plane. This fact shews at once that, if e be the inclination of the plane to the horizon, the reaction (which must balance the weight of the body) acts at angle ɛ to the normal to the plane in the opposite direction to that in which the body is on the point of sliding.

Hence, if the body be on the point of sliding up the plane under the action of a force (P) along the plane, the reaction (R) will act as in fig. 40, where NOR between its direction and the normal Ε.

Let ML (fig. 40) drawn parallel to OP meet RO produced in L. Then the sides of the triangle OLM, being parallel to the directions of the three forces P, W, R, which keep O at rest, will be proportional to their magnitudes; i.e.

1850.

sin OLM'

sin 2€

COS E

2 sin ɛ.

P = 2 W. sin ɛ.

(A). Find the ratio of the power to the weight necessary for equilibrium on an inclined plane, when the power acts along the plane.

(B). If the inclined plane be the upper surface of a wedge whose under surface rests on a smooth horizontal table, find the horizontal force which must act on the wedge to keep it at rest.

From the equilibrium of the particle O (fig. 41) we have, by the same method as is pursued in (4),

From the equilibrium of the wedge, observing that the effect of the particle resting on the wedge is to impress upon it a force R perpendicular to the slant side, we have, by equating horizontal

1850. (4). Find the ratio of the power to the weight necessary for equilibrium on the wheel and axle.

(B). If the axis about which the machine turns coincide with that of the axle, but not with the axis of the wheel, find the greatest and least ratios of the power and weight necessary for equilibrium, neglecting the weight of the machine.

Let 0 (fig. 42) be the axis of the axle, O' of the wheel.

1849.

(A). Find the relation between P and W in the system of pullies in which the same string passes round all the pullies.

(B). A triangular plane ABC is kept in equilibrium by three systems of pullies of the above kind, each having one block fastened to a fixed external point and the other attached to an angular point of the triangle by a string whose direction bisects the angle. The same string passes round all the pullies and is solicited by a certain force. Shew that the numbers of the strings between the pullies are as cos 4: cos B: cos C.

If n be the number of the parallel strings in (A),

Let n1, nã, nã, be the numbers of strings between the blocks of the pullies at A, B, C, respectively. Then, by the above formula, the tensions at A, B, C, which keep the triangle at rest, will be n‚P, n ̧P, n,P; P being the force which acts upon

the string passing round all the pullies. The directions of these three tensions, by hypothesis, bisect the angles of the triangle, and therefore meet in a point. Let them meet in O (fig. 43). Then, supposing O to be a point rigidly connected with the triangular plane, we may regard the tensions as acting upon it; and we have

nPnP: nP:: sin BOC: sin AOC: sin AOB,

or

:: sin (B+C): sin (A+C): sin(A+B),

n1n ng cos 4: cos B: cos C.

2

3

( 56

DYNAMICS.

1851. (A). Write down the laws of motion; giving any illustrations you please for the sake of explanation.

(B). If a weight of 10 lbs. be placed upon a plane which is made to descend with a uniform acceleration of 10 feet per second, what is the pressure upon the plane? Let R be the pressure on the plane. The moving force upon the given weight

10 lbs.

R,

= (mass) × (acceleration per second), by the third law of motion,

1850.

(4). Explain clearly on what conventions with respect to units the equation P= Mf is true, where f expresses the accelerating effect of a force whose statical measure is P.

(B). A body weighing 10 lbs. is moved by a constant force which generates in a second a velocity of 1 foot per second; find what weight the force would support.