(8) A weight is placed upon a smooth inclined plane: shew that it is impossible for a force, acting at right angles to the plane, to produce equilibrium.

(9) A wheel, moveable about its centre in a vertical plane, has three weights equal to 1, 2, 3, at three equidistant points in the circumference: as it turns round, find the greatest moment of the weights about the centre.

The moment will be greatest whenever the weight 1 is at its greatest distance from a diameter of the wheel which is inclined to the horizon at an angle of 60°.

(10) A rod is supported horizontally by two props, at given distances from its centre of gravity: find the pressure on each. If a, b, represent the two distances, P, Q, the corresponding pressures, and W the weight of the rod,

(11) ABC is a heavy triangle, of weight 9 lbs. AD bisects BC in D, and DA is produced to F, AF being equal to AD. If AF be a rod without weight, rigidly connected with the triangle ABC, what weight must be suspended at F to balance the triangle about a fulcrum at A?

A weight of 6 pounds.

(12) A uniform rod, attached at its lower end to a smooth hinge on a horizontal plane, leans with its upper end against an inclined plane which rests on the horizontal plane: determine the horizontal force which must be applied to the inclined plane to prevent its sliding.

If a be the inclination of the plane and ẞ of the rod to the horizon, and W the weight of the rod, the required horizontal force is equal to

I W
cot a + tan B

(13) If a set of forces, acting on a body along the sides, taken in order, of a plane polygon, be proportional to the sides, shew that their tendency to turn the body about an axis, perpendicular to the plane of the polygon, is the same through whatever point of the plane the axis passes.

-1

(14) An inclined plane, without weight, with a rough base and a smooth inclined face, is set upon a rough horizontal plane: shew that, if the angle of the inclined plane be less than tanμ, μ being the coefficient of friction between the surfaces in contact, no force, applied to the inclined face, will be able to move the inclined plane.

(15) A string ABCDEP, fig. (102), is attached to the centre A of a pully, the radius of which is r; it then passes over a fixed point B, and under the pully, which it touches in the points C and D; it afterwards passes over a fixed point E, and has a weight P attached to its extremity; BE is horizontal and equal tor, and DE is vertical: supposing the system to be in equilibrium, find the weight of the pully and the distance AB.

5

5 2

The weight of the pully == P and AB =

8r

3√7

(16) An endless string supports a system of equal heavy pullies, the highest one of which is fixed, the string passing round every pully and crossing itself between each. If a, ß, y,... be the inclinations to the vertical of the successive rectilinear portions of the string, prove that cos a, cos B, cosy,... are in arithmetical progression.

(17) Three uniform rods, rigidly connected in the form of a triangle, rest on a smooth sphere of radius r: prove that the inclination of the plane of the triangle to the horizon is equal to tan-1 where d is the distance between the centres of the circles inscribed in the triangle itself and in the triangle formed by joining the middle points of the rods.

(18) A uniform rod is held at a given inclination to a rough horizontal table, by a string attached to one of its ends, the other end resting on the table: find the greatest angle at which the string can be inclined to the vertical, without causing the end of the rod to slide along the table.

If a be the inclination of the rod to the table, and tan e the coefficient of friction, the required angle is equal to

cot (cote+2 tan a).

(19) A picture is hung up against a rough vertical wall by a string fastened to a point in its back, so that the picture inclines forwards: apply the principle of the triangle of forces to find the inclination of the string to the wall, when its tension is the least possible.

If tane be the coefficient of friction, the required inclination is equal to e.

(20) A winch handle, by describing a circle, turns a small cog-wheel, which turns the wheel of a Wheel and Axle; the weight to be raised is attached to a single moveable pully; one portion of the string supporting the pully is wound round the axle, and the other is tied to a fixed bar, the strings being parallel; find the mechanical advantage in terms of the radii of the circle described by the winch handle, and of the axle, and of the number of teeth in the cog-wheel, and in the wheel of the Wheel and Axle.

If r be the radius of the circle described by the winch handle, r' the radius of the axle, n the number of teeth in the wheel of the Wheel and Axle, and n' the number in the cog-wheel, the mechanical advantage is equal to

DYNAMICS.

CHAPTER I.

FREE RECTILINEAR MOTION.

SECT. 1. Uniform motion.

(1) Two candles, which will burn for four and six hours respectively, are placed in candlesticks a foot high and a foot apart, and are lighted at the same moment. The shadow of the shorter is received on the table on which the candlesticks stand; that of the extremity of the longer on a vertical wall, ten feet distant from it, and perpendicular to the plane of the candles. Supposing each candle to be originally a foot long, to find the velocity of the extremity of the shadow of the longer: to find also the mean velocity of the extremity of the shadow of the shorter, during the last hour in which it is burning.

Let AB, A'B', fig. (103), be the candlesticks, BC, B'C', the lengths of the candles at the end of t hours from the moment of lighting, and K, L, the extremities of the shadows of the longer and shorter candles respectively.

Let BC=x, B'C' =y, AL=a, HK=b. Then

Hence, substituting for x, y, their respective values in terms of t, we have

The equation (3) shews that a = 5 when t =

=

(3).

(4).

3, and that a = 3 when 4: hence the extremity of the shadow of the shorter candle describes 2 feet in the last hour: its mean velocity is therefore, during the last hour, 2 feet per hour.

Again, the equation (4) shews that the extremity of the shadow of the longer candle has a uniform velocity of 8 inches per hour.

(2) One body moves through 40 feet in 3 seconds, and another through 25 yards in 6 minutes: compare their velocities.

The velocity of the former body is 64 times as great as that of the latter.

(3) What is the ratio of the velocity of light to that of a cannon ball, which issues from a gun with a velocity of 1500 feet per second, light passing from the Sun to the Earth, a distance of ninety-five millions of miles, in 83 minutes?

The required ratio is 668800.