will balance a weight of 130 lbs. The power being supposed to act in the axis of the

rope.

109. The radius of a wheel being 3 feet, and of the axle 3 inches; find what weight will be supported by a power of 120 lbs. The thickness of the rope coiled round the axle being I inch.

110. There are two wheels, whose respective diameters are 5 feet and 4 feet, on the same axle; the diameter of the axle being 20 inches. What weight on the axle would be supported by forces equal to 48 lbs. and 50 lbs. on the larger and smaller wheels respectively?

III. At what angle must the strings of a single moveable pulley be inclined to each other, in order that P may equal W?

112. How many pulleys (supposed to be without weight) must there be in the first system, in order that I lb. may support a weight of 128 lbs.?

113. In the first system there will be equilibrium, if the power, the weight, and each pulley are all equally heavy.

114. In the first system of 6 moveable pulleys, find the ratio that the weight of each pulley must bear to the power, in order that the latter may just be balanced by the weight of the pulleys alone.

115. In the second system of pulleys, if there be 10 strings at the lower block; find what power will support a weight of 1000 lbs. 116. In the third system of 6 pulleys (supposed to be without weight); find what weight will be supported by a power of 12 lbs. 117. In the third system of 8 pulleys, find the ratio that the weight of each pulley must bear to the weight supported, in order that the latter may just be supported by the weight of the pulleys alone.

118. What force is necessary to support a weight of 50 lbs. on a plane inclined at an angle of 15° to the horizon; the force acting horizontally?

119. If, on an inclined plane, the pressure, force and weight be as the numbers 4, 5, and 7; find the inclination of the plane to the horizon, and the inclination of the force's direction to the plane.

120. If the weight, power, and pressure on an inclined plane be respectively as the numbers 25, 16, and 10; find the inclination of the plane, and the inclination of the power's direction with the plane.

121. A weight W is just supported on an inclined plane by a force P, acting by means of a wheel and axle placed at the top, so that the string attached to the weight is parallel to the plane. Given R and r the radii of the wheel and of the axle; find the plane's inclination to the horizon.

122. What force must be exerted to sustain a ton weight on a screw, the thread of which makes 150 turns in the height of 12 inches; the length of the arm being 6 feet?

123. Find the weight that can be sustained by a power of 1 lb., acting at the distance of 3 yards from the axis of the screw; the distance between two contiguous threads being I inch.

124. What must be the length of a lever, at whose extremity a force of 1 lb. will support a weight of 1000 lbs. on a screw; the distance between two contiguous threads being inch?

FRICTION.

3

When a body rests on any surface, the friction =μR, where R is the normal pressure at that point; μ being constant and called the coefficient of friction.

This friction is therefore to be accounted for as one of the forces acting on the body, and is considered to act in a direction opposite to that in which the body is on the point of moving. But, as it is a force of a peculiar nature, not causing motion, but only preventing other forces from causing motion, there may be a great many conditions of equilibrium, with either of which, the friction is sufficient to present sliding. But there will be, in general, limits to those conditions, with one of which, motion is supposed to be on the point of taking place in one direction, and with the other, in the opposite direction. One of these conditions may always be found from the other, by changing the sign of μ; which is equivalent to reckoning the friction in the opposite direction.

The coefficient of friction is, for practical purposes, often expressed in terms of the inclination of the plane, on which the body being placed will be just on the point of descending. The friction R is thus a force acting up the plane, which keeps the body at rest, and therefore u R: R= tan a : 1, or μ=tan a, a being the plane's inclination.

μ

125. A given weight W is sustained on a rough plane, whose angle of inclination to the horizon is a, by a power P, inclined at an angle ẞ to the plane; μ being the coefficient of friction. Find between what limits P must lie.

126. Determine the least force which will drag a weight of 50 lbs. along a rough horizontal plane, the friction being such as would just prevent the body from sliding down a plane of inclination 30°. Find also at what angle this least force must be inclined to the horizon.

127. A body is supported on a rough plane by a force equal to its own weight, and is just on the point of sliding down the plane. Required the sum of the angles at which the plane is inclined to the horizon and to the force's direction.

128. An isosceles triangle, whose base is to one of its equal sides as I: 7, is placed with its base on an inclined plane; and it is found that when the body begins to slide, it also begins to roll over. Find the coefficient of friction.

129. An uniform rectangular beam AB, of weight W, is supported against a rough wall CD by a string AC passing over a pulley at C, to which a weight P is attached. If the inclination of the plane which measures the friction be 30°; and if the centre of gravity of the beam be at G the middle point of ADB; find whether P must be greater or less than W.

Resolve vertically, and horizontally, and take moments about D.

130. An uniform beam, of weight W, leans against a vertical wall, and has its lower end resting on a horizontal plane. If μ and ' be the coefficients of friction of the wall and of the plane respectively; find the value of the inclination of the beam to the horizon, when motion is just on the point of taking place. Find also the pressures on the wall and on the plane.

Resolve vertically, and horizontally, and take moments about the lower end of the beam.

131. A ladder rests against a vertical wall, to which it is inclined at an angle of 45°; the coefficients of friction of the wall and of the horizontal plane being respectively and;

and the centre of gra

A man, whose weight. find to what height he

3 vity of the ladder being at its middle round. = half the weight of the ladder, ascends it; will go before the ladder begins to slide. Resolve, and take moments as in the last example.

132. A hemisphere rests between a vertical wall and a horizontal plane; and being the coefficients of friction of the wall and plane respectively. Find the inclination of the base of the hemisphere to the horizontal plane, in the limiting position of equilibrium.

Resolve vertically, and horizontally, and take moments about the centre of the sphere.

134

I.

DYNAMICS.

THE COLLISION OR IMPACT OF BODIES.

General Formulæ. Let A, B be the masses of the two bodies that impinge, one on the other; a, b their velocities before impact and measured in the same direction; u, v the velocities after impact; the common elasticity of the two bodies: then will Aa+Bb−ɛB(a−b),

u=

A+B
Aa+Bb+ɛA(a−b).

v=

A+B

N.B. When the bodies are inelastic, ɛ=0; when perfectly elastic, = 1.

1. A and B weigh 12 lbs. and 7 lbs. respectively, and move in the same direction with velocities of 8 feet and 5 feet in a second; find the common velocity after impact; also the velocities lost by A and gained by B respectively.

2. A, moving with a velocity (11) impinges upon B moving in the opposite direction with a velocity (5); and by the collision A loses one-third of its momentum; what are the relative magnitudes of A and B ?

3. A, weighing 8 lbs., impinges upon B weighing 5 lbs. and moving in A's direction with a velocity of 9 feet in 1"; by collision B's velocity is trebled; what was A's velocity before impact?

4. The centres of two balls, A and B, move along the same straight line with velocities a and b. Find the velocity of each after impact when 6A=5B, a is 7 feet in 1", 4a+5b=0, and ɛ==•

2

3

5. A and B are two perfectly hard balls meeting in opposite directions; A is three times as large as B, but moves with a velocity of 12 feet in 1", which is only 3rds of B's velocity; what is the common velocity after impact?

:

3

6. A B as 3 2, and the velocity of A: velocity of B as 5: 4; they are perfectly hard bodies and move before impact in the same direction. What is the proportion of A's velocity lost by A and gained by B?

7. A and B are perfectly elastic and in the ratio of 4: 3; they are moving in the same direction with velocities as 5:4; what is the ratio of the velocities of A and B after impact?

8. A and B are perfectly elastic; they are moving in opposite directions; A is treble of B, but B's velocity is double that of A ; determine the motions after impact.

9. There is a row of perfectly elastic bodies in an increasing geometrical progression whose common ratio is 3, and placed contiguous to each other; the first impinges upon the second, which transmits its velocity to the third, and so on; the last body moves off with I th of the velocity of the first body; what was the number of bodies? 10. A(=3B) impinges upon B at rest; A's velocity after impact is 3ths of its velocity before impact; required the value of # which measures the elasticity.

64

5

I

E

II. At what angle must a body, whose elasticity is, be incident

3

on a perfectly hard plane that the angle between the directions before impact and after, may be a right angle?

12. A and B are two balls of given elasticity; what must be the magnitude of a third ball, that the velocity communicated from A to B by the intervention of this ball may equal that communicated immediately from A to B? Determine also the limits within which this problem is possible.

13. If A communicate velocity to B through a number of other bodies which are geometric means between A and B, find the limit to which the velocity of B will continually approach when the number of means is continually increased, the bodies being perfectly elastic.

14. ABD is a semicircle, AB the diameter, DC the sine of BD. An inelastic body moving uniformly along AD impinges on the plane CD; prove that it will move uniformly along DC, and that the time along DC will vary as tan BD.

15. A ball, whose elasticity is ɛ, projected from a given point in the circumference of a circle, after being reflected from it twice returns to the given point. Required the direction of projection.

16. The sides AB, CD of a billiard table are parallel; and an imperfectly elastic ball struck from a point C in one side, impinges at E in the other, and is reflected to D in the first side. Show that the time along CE : time along ED :: ɛ : I.

17. A row of four balls A, B, C, D of perfect elasticity is placed in a straight line. Required the ratio of their masses, that the momentum of A may, after impact, be equally divided among the four; B, C, D being originally at rest.

Find the result also when there are n balls.