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18. The tangent + cotangent of an angle equal twice the cosecant of twice the angle.
19. Having given two sides of a triangle and the included angle, shew how the other parts may be found.
20. A on foot, and B on horseback, are travelling together towards the same town; A takes a foot path leaving the road at a given angle, B goes on till he comes to a cross road. They arrive at the same instant. Given their rates of motion and greatest distance of separation, find the distance travelled by each, and the species and dimensions of the included area.
21. If p, q, r, s, be the coefficients of an equation whose roots are the tangents of A, B, C, 8c. Then the tangent of A + B + C+ &c. =
p -1 + t - &c.
- 9+8 - &c. 22. A person wishing to know the height of a spire due south of him, observes a small cloud pass behind it, the wind blowing southwest. Soon after the cloud passed over the Moon. He then measures his distance from the foot of the spire; and on return home calculates the Moon's altitude and angular distance from the south at the time of observation. Shew how from these data he may determine the height of the spire.
1. When forces keep each other in equilibrium round a fixed point, the sum of all their moments is = 0; those being reckoned negative which tend to turn the system in the opposite direction.
2. Find the resultant of any number of forces in the same plane acting on a point. Apply the formulæ to the following example:
AB, AC, AD, are three lines making angles of 120° with each other; the point A is acted on by pulling forces in AB and AC, which are as 3 and 4, and by a pushing force DA, which is as 5. Find the force which will keep it at rest.
3. A string fastened at A and passing over a fixed pulley B, has
a known weight W hung by a knot at C; find what weight must be appended at P, that CB may be horizontal.
4. A weight Q hanging freely, supports an equal weight P upon an inclined plane, by means of a string passing over a pulley below the plane : find the position of equilibrium.
5. When a body is sustained upon a curve whose co-ordinates are x and y, by any forces whose components in those directions are X and Y, shew that
Xdx + Ydy = 0. Apply the formula to find the position of equilibrium when a weight Q hanging freely, supports a weight P upon a parabola whose axis is horizontal, by means of a string passing over the focus.
6. Find the centre of gravity of any number of points in the same plane.
7. The sum of the squares of the distances of three equal bodies from each other, is three times the sum of the squares of their distances from their common centre of gravity.
8. Prove the differential expression for the centre of gravity of any solid of revolution; and find the centre of gravity of a hemisphere.
9. ABCD is a quadrilateral figure, of which the two shorter sides AB, BC are equal, as also the two longer AD, DC; and the angle ABC is a right angle: what is the greatest length of the side AD, that the figure may stand on the base AB on a horizontal plane without oversetting?
10. Given a bent lever with arms of uniform thickness, moveable in a vertical plane about the angular point: find the positions in which it will rest.
11. A given beam considered as a line is supported on two given inclined planes : find the position of equilibrium.
12. Given the pressure upon one of the four legs of a rectangular table of known weight; find the pressures of the other three. Shew that without this datum the problem is indeterminate.
13. ABC is a right-angled isosceles triangle, and three equal forces act in the lines AB, BC, CA. At what point of the plane ABC, produced if necessary, must a force be applied to keep it at rest, and what must be its magnitude and direction?
14. A beam BC hangs by a string AB from a fixed point A, with its lower extremity C upon a horizontal plane: find the position in which it will rest. Also find the horizontal force which must be applied at C to retain it in a given position.
15. A false balance has one of its arms exceeding the other by 1
of the shorter. It is used, the weight being put as often in one scale as the other. What is the shopkeeper's gain or loss per cent. ?
16. In an arch which is jo equilibrium, the weights of the vous
soirs are as the differences of the tangents of the angles which their joints make with the vertical.
1. A Bow is drawn by a force of 50 lbs; the weight of the arrow being rolb, compare the force of gravity with the initial accelerating force which the string exerts upon the arrow, when it is let go; neglecting the inertia of the bow.
2. If a, b, be the velocities of two bodies A, B before their direct impact; u, v the velocities after, a and B the vclocities gained and lost respectively, and e the fraction which measures the elas. ticity;
Ite 3. A and B are two given points in the diameter of a circle : find in what direction a perfectly-elastic body must be projected from A, so that after reflection at the circle it may strike B. 4. Prove that if a body be accelerated by a constant force
v = ft and s = 1.ft. 5. Find the velocity and direction with which a body must be projected from a given point, that it may hit two other given points in the same vertical plane.
6. AB is the vertical diameter of a circle: a perfectly elastic body descends down the chord AC; and being reflected by the plane BC, describes its path as a projectile. Shew that this path strikes the circle at the opposite extremity of the diameter CD.
7. Find the equation to the cycloid ; and shew that in the same cycloid the oscillations are isochronous.
1. The pressure of a fluid against any surface in a direction perpendicular to it, varies as the area or the surface multiplied
into the depth of its centre of gravity below the surface of the fuid.
2. A hollow cone without a bottom stands on a horizontal plane, and water is poured in at the vertex. The weight of the cone being given, how far may it be filled so as not to run out below?
3. What must be the magnitude and point of application of a single force that will support a sluice-gate in the shape of an inverted parabola ?
4. Find the specific gravity of a body which is lighter than the Auid in which it is weighed.
5. If the specific gravity of air be called m, that of water being 1, and if W be the weight of any body in air, and W' its weight in water, its weight in vacuo will be,
6. Three globes of the same diameter and of given specific gravities, are placed in the same straight line. How must they be disposed that they may balance on the same point of the line in vacuo and in water ?
7. If a homogeneous hemisphere, foating in a Auid, be slightly inclined from the position of equilibrium ; shew that the moment of the fluid to restore it to that position, is not affected by placing any additional weight at its centre.
8. A regular tetrahedron moves with its vertex forwards, in a direction perpendicular to its base. Compare the resistance on the oblique planes with that on the base
9. If the particles of an elastic fluid repel each other with forces varying inversely as the fourth power of their distances, the compressive force on any portion varies as (density).
10. Explain the method of measuring altitudes by means of the barometer and thermometer.
11. Two barometers, whose tubes are each 1 inches long, being imperfectly filled with mercury, are observed to stand at the heights h and h', on one day, and k and k' on another. Find the quantity of air left in each, reducing it to the density when the mercury is at the standard altitude of 30 inches, and supposing the temperature invariable.
12. Construct a common forcing-pump; and shew what is the force requisite to force the piston down.
13. In the common sucking-pump, given the lowest point to which the piston descends, find the length of the stroke ihat the pump may work.
14. A cylinder which floats upright in a fluid, is pressed down below the position of equilibrium : when it is left to itself, find the time of its vertical oscillations, neglecting the resistance.
15. A vessel generated by the revolution of a portion of a semi
hyperbola round its conjugate axis, is emptied by an orifice at the centre of the hyperbola: find the time.
16. A close vessel is filled with air n times the density of atmospheric air. A small orifice being made, through which the air rushes into a vacuum, find the time elapsed when the density diminished one half.
17. A tube of uniform diameter consists of two vertical legs connected by a horizontal branch. When it is made to revolve with a given velocity round one of its vertical legs as an axis, find the height to which the water will rise in the other.
18. Let a spherical body descend in a fluid from rest; having given the diameter of the sphere and its specific gravity relatively to that of the fluid, it is required to assign the time in which the sphere describes any given space.
119. If the density of a medium vary inversely as the distance from a center, and the centripetal force inversely as any power of the distance from the same centre, a body may describe a logarithmic spiral about that point.
20. If the resistance on a body which oscillates small arts in a fluid vary as the nth power of the velocity, the difference of the arcs of descent and ascent will vary as the need power of the whole
1. Given the distance of the focus of incident rays from the centre of a given spherical resector; find the distance of the geometrical focus of reflected rays from the centre, when they are incident nearly perpendicularly.
2. Given the position of an object placed between two plane reAlectors inclined at a given angle; find the total number of images, and apply it to the case where the angle of inclination is equal to 11°. 15'.
3. A straight line is placed before a concave spherical reflector, at the distance of one-third of its radius from the surface ; find the dimensions of the curve formed by its image, the radius of reflector being 9 inches.