Ex. 7.

16. A rigid rod SA without weight, 5 feet long, passes through the centres of three spheres C, B, A whose radii are 2, 3, 5 inches respectively, so that SC 40 in., CB= 12 in., BA=8 in., S being the point of suspension; find the time of an oscillation.

=

17. Find at what point of the rod of a perfect pendulum, must be fixed a given weight of indefinitely small volume so that the pendulum may vibrate in the shortest time possible.

18. If the interior of two circles, which touch internally, be taken away, and the remaining area oscillate about an axis in its own plane find which is a tangent at the point common to the two circles; the centre of oscillation.

19. Compare the times in which a circular plate will vibrate about a horizontal tangent, and about a horizontal axis through the point of contact at right angles to the tangent.

20. Find the isosceles triangle of a given area, which, vibrating about an axis passing through its vertex perpendicular to its plane, shall oscillate in the least time possible.

21. A sector of a circle revolves about an axis perpendicular to its plane, and passing through the centre of the circle; find the angle of the sector when the length of the isochronous simple pendulum equals one-half the length of the arc.

22. Prove that a right cone, whether suspended at its vertex or by the diameter of its base, will oscillate in equal times; the height of cone being equal to the radius of its base.

23. Determine the ratio of the diameter of the base to the altitude of a cone, so that the centre of oscillation, when the cone is suspended by the vertex, may be in the centre of the base.

24. Find the dimensions of a cone of given volume, which, being suspended by the vertex, will oscillate as many times in a minute as there are inches in the length of its axis.

25. A cylindrical rod of given length oscillates seconds, when suspended from one extremity; at what point must it be suspended to oscillate once in n seconds?

26. A pendulum consists of a rigid rod OA without weight, and a sphere of which the centre is A and radius r; to determine the point A' in the line OA at which the centre of another sphere of radius must be fixed in order that the time of oscillation of the system of 2 spheres may be the least possible.

27. Two straight rods, equal in length, are suspended by their extremities, one being of uniform density, and the density of the other varying as the nth power of the distance from the axis of suspension; the times of their small oscillations are found to be as 5: 6; required the value of n.

28. A bent lever, whose arms are a, b, and inclination to one another, makes small oscillations in its own plane about the angular point; find the centre of oscillation.

Ex. 7.

29. A uniform rod of length a is bent into the form of a cycloid, and oscillates about a horizontal line joining its extremities; find the length of the isochronous pendulum.

30. If two particles whose weights are as 2 : 3 oscillate, one in a semicircle, and the other in a cycloid; show that the whole tensions of the two strings, at any given inclination of them to the horizon, are equal; the motion in both cases beginning from the highest point.

Ex. 8.

III. D'ALEMBERT'S PRINCIPLE.

1. Two heavy particles P, P' are attached to a rigid imponderable rod APP', which is oscillating in a vertical plane about a fixed point in its extremity A; determine the motion.

2. Two particles, attached to the extremities of a fine inextensible thread, are placed upon two inclined planes having a common vertex; determine the motion of the particles and the tension of the thread at any time.

3. One body m draws up another m' on the wheel and axle; determine the motion of the weights and the tensions of the strings. 4. A body P, draws another body Q, over a fixed pulley AB; determine the motion, and tensions of the string.

5. A square is capable of revolving about a side; find where it must be struck perpendicularly that there may be no impulse upon

the axis.

6. Suppose a cylinder, that weighs 100 lb., to revolve upon a horizontal axis, and to be set in motion by a weight P of 15 lb. attached to a string which is coiled round the surface of the cylinder; find the space through which the weight descends in 5 seconds.

7. If P in the preceding Ex. be 20 lb. and descend through 75 feet in 3 seconds; what is the weight of the cylinder?

8. A sphere C, of radius 3 feet and weight 500 lb., is put in motion by a weight P of 20 lb. by means of a string going over a wheel whose radius is 6 inches; in what time will P descend through 50 feet, and what velocity will it then have acquired?

9. A paraboloid whose weight W is 200 lb. and radius of base 20 inches, is made to revolve about its axis, which is horizontal, by means of a weight P of 15 lb. acting by a cord that passes over a wheel of one foot diameter on the same axis; after P has descended for 10 seconds, it is removed, and the paraboloid is left to revolve uniformly with the velocity acquired; find the velocity of the centre of gyration of the paraboloid, and the number of revolutions it will perform in one minute.

Ex. 8.

10. Two weights of 5 lb. and 3 lb. hang over a fixed pulley, whose weight is 12 oz.; find the time of either weight moving through a space of 30 feet.

11. What weight could be raised through a space of 30 feet in 6 seconds by a weight of 50 lb. acting by means of a string going round a fixed and a moveable pulley, the weight of each pulley being 1 lb.?

12. A weight of 500 lb. is raised by a rope wound round an axle whose radius is 6 inches; the weight of the wheel and axle is 80 lb., and the distance of their centre of gyration from the axis of rotation is 3 feet; find the radius and weight of the wheel, so that another weight of 100 lb., acting at its circumference, may make the 500 lb. ascend through a space of 10 feet in 5 sec.; also find the pressure upon the axis during the motion.

13. A hemisphere oscillates about a horizontal axis, which coincides with a diameter of the base; if the base be at first vertical, find the ratio of the greatest pressure on the axis to the weight e the hemisphere.

e

HYDROSTATICS.

PRESSURE ON SURFACES.

I. Let A be the area of any surface immersed in a fluid, a the depth of the centre of gravity of the surface A below the surface of the fluid, the density of the fluid, and P the normal pressure on A; then

Р

II. The vertical pressure on A is equal to the weight of the fluid incumbent on A.

Ex. 1.

1. An equilateral triangle is immersed in a fluid so that one side is vertical; compare the pressures on the three sides.

2. Two equal isosceles triangles are just immersed vertically in a tquid, one with its base, the other with its vertex downwards; find the ratio of the pressures.

3. An isosceles triangle with its base downwards is just immersed vertically in a fluid; divide the triangle by a line parallel to the base so that the pressures on the upper and lower parts may be as I:7.

4. A triangle, the area of which is A, being immersed in a fluid with its angular points at depths h, k, l below the surface of the fluid; it is required to find the pressure on the triangle.

5. Two equal squares are just immersed vertically in a fluid, one with a side, the other with a diagonal vertical; find the ratio of the

pressures.

6. Two squares, whose sides are 9 and 5 inches respectively, are immersed vertically in a fluid, their sides being parallel to its surface. The first square has its upper side at a depth of 4 inches beneath the surface; find the depth to which the second square must be sunk, so that the pressure on it may be 3 times that on

the first.

7. The sides of a rectangle immersed vertically in a fluid are 9 and 14, the shorter side being coincident with the surface. From one of the angles at the surface draw a straight line to the base, dividing the rectangle into two parts, such that the pressures on them may be in the ratio of 5: 3.

8. A rectangle, whose sides are 20 and 7, is immersed vertically in a fluid, with its shorter side coincident with the surface; divide the rectangle into 5 parts by horizontal lines, so that the pressures on each part may be equal.

9. A parallelogram, of which the diagonals AC, BD intersect in

Ex. 1.

O, is immersed in a fluid, so that AB is in the surface of the fluid; compare the pressures L, M, Non the triangles AOB, BOC, COD. 10. A circle whose radius is 6, is just immersed vertically in a fluid; find the radius of a circle touching the former internally at the surface of the fluid, so that the pressures on the smaller circle and frustum may be as 5 : 4.

11. If a circle be inscribed in a square, and another square be inscribed in the circle, and the whole figure be then just immersed vertically in a fluid, so that an angular point of the greater square may coincide with the surface; compare the pressures on the squares and circle.

12. A rectangle is described about a parabola, and the whole figure is immersed vertically in a fluid, so that the vertex coincides with the surface of the fluid; compare the pressures on the parabola and rectangle.

13. A parabola is immersed in a fluid with its axis vertical, and vertex coincident with the surface; divide the parabola by a hori-✔ zontal line into two parts, so that the pressures on them may be ás

m: n.

14. A parabola with its axis vertical, has its vertex coincident with the surface of the fluid in which it is immersed; divide the parabola by horizontal lines into four parts, so that the pressures on them may be equal.

15. If a cubical vessel be filled with fluid and rest on one of its sides; compare the vertical and lateral pressures.

16. A cubical vessel filled with fluid is held with one of its diagonals vertical; compare the pressures on the sides.

17. If a cubical vessel be filled, half with mercury and half with water; compare the pressure on the sides with the pressure on the base, which is horizontal.

18. A side of the base of a square pyramid is 10 inches, the altitude is 22 inches; if the pyramid be filled with water, compare the pressure on the base with the pressure on each side, and with the weight of the water.

19. If two spheres whose radii are as 3 : 5, be just immersed in a fluid; compare the pressures on them.

20. If a given sphere be just immersed in a fluid; compare the pressure on the sphere with the weight of the fluid displaced.

21. If the density of mercury be 13.568 times that of water; it is required to compare the pressure on the internal surface of a sphere filled with water, with the weight of a sphere of mercury of the same radius.