7. Two forces are in the ratio of 3: 2; find the angle between their directions, when the resultant is a mean proportional between them.

8. Ten men of equal strength wishing to pull down a tree of given height, and at the same time to avoid all danger from its fall, fix two ropes at its top, one of which reaches the ground making an angle of 60° with the tree, and the other 45o. If four men pull at the former towards the south, and six at the latter towards the south-east, towards what point of the compass will the tree fall?

9. Given the resultant of two forces, their sum, and the angle between their directions; find the forces.

10.

Three forces acting upon a point, keep it at rest; and they are in the ratios of √3 +1: √6 : 2. Find the angles at which they are respectively inclined to each other.

11. A small ring is attached to one end of a string of given length, the other end of which is fixed to a given point. Another string is fixed to a given point in the same horizontal line as the former, and this string passing through the ring supports a weight; find the position of the ring.

12. Four forces represented by 1, 2, 3, and 4, act in the same plane on a point. The directions of the first and third are at right angles to each other; and so are the directions of the second and fourth; and the second is inclined at an angle of 60° to the first. Find the magnitude and direction of the resultant.

13. A circular hoop is supported in a horizontal position, and three weights of 4, 5, and 6 pounds respectively are suspended over its circumference by three strings knotted together at the centre of the hoop. Find the angles between the strings when there is equilibrium.

14. Two equal weights are supported by a string which passes over three tacks, forming a vertical isosceles triangle of which the base is horizontal and the angle opposite to the base 120o. Find the pressure on each of the tacks.

15. Three forces represented by the numbers 3, 5, 9 cannot under any circumstances produce equilibrium upon a point.

16. If three forces in equilibrium upon a point are represented by the numbers 3, 4, 5, respectively, two of them are perpendicular to each other.

17.

The resultant and sum of two forces being given, and also the angle which one of them makes with the resultant; it is required to determine the forces and the angle at which they act.

PRINCIPLE OF THE LEVER.

1. AC, CB are the equal arms of a straight lever whose fulcrum is C: to Ca heavy arm CD is fixed perpendicular to AB. Prove that if a weight be suspended to the extremity A, and the system be in equilibrium, the tangent of the inclination of CD to the vertical will be proportional to the weight.

2. Four weights, 1, 3, 7, 5 are placed at equal distances on a straight lever. Determine the fulcrum.

3. There are n weights, W1, Wa...... W, in geometrical progression, and W1 placed at A, one extremity of a lever, balances W2 placed at B, the other extremity. Prove that a weight equal to the first n 1 weights, if placed at A, will balance a weight equal to the last n − 1, if placed at B.

2

4. The lever AC (without weight) turning about the fulcrum C, has two given weights W, W' suspended from the extremity A and the middle point B respectively, and is kept at rest by the given weight P acting at A by means of a string passing over a tack at D: CD is horizontal and equal to AC: find the position of equilibrium.

5. A uniform bent lever, when supported at the angle, rests with the shorter arm horizontal. If the shorter arm were twice as long, it would rest with the other horizontal. Compare the lengths of the arms, and find the angle between them.

6. A straight lever of uniform thickness, the length and weight of which are given, has two weights P and Q attached to its extremities, and is sustained partly by a fulcrum at a given point, and partly by another fulcrum, on which it presses with a given force; required the position of this latter fulcrum.

7. A uniform bent lever ABC, containing a given angle at B, and having its arms of given weights and lengths, hangs freely by the extremity A: find the position of equilibrium.

8. A straight lever is sustained on a fulcrum at the middle point, and is kept at rest by two given weights; where must they be placed, in order that the distance of the one from the fulcrum may equal the distance of the other from the extremity? And where must the fulcrum be placed, if the position of the weights be reversed?

9. A uniform rod of given weight and length suspended at a given point, is drawn out of the vertical position by a given force, acting upon its lower extremity by means of a string, which passes over a peg at a given point in the same horizontal line with the axis of suspension; find the angle through which the rod is drawn.

10. Two weights, 3 and 4, balance on the extremities of a lever 4 feet long; find the fulcrum.

11. A uniform beam of given weight and length is moveable about its middle point; a given weight is hung by a string to one end; find to what extent the other end of the beam must be lengthened in order that there may be equilibrium.

12. In question 8, page 66, determine the moment of the force which is effective in pulling down the tree.

13. A bar weighs a oz. per inch. Find its length when a given weight na, suspended at one end, keeps it in equilibrium about a fulcrum at a distance of b inches from the other end.

14. A beam, 30 feet long, balances about a point at onethird of its length from the thicker end; but when a weight

of 10 lbs. is suspended from the smaller end, the fulcrum must be moved 2 feet towards it in order to maintain equilibrium. Find the weight of the beam.

15. At what point of a tree must a rope of given length be fixed, so that a man endeavouring to pull down the tree may have the greatest advantage?

16. A uniform beam 18 feet long, rests in equilibrium upon a fulcrum 2 feet from one end, having a weight of 5 lbs. at the end furthest from the fulcrum and one of 110 lbs. at the other. Find the weight of the beam.

17*. Three equal rods without weight are attached together, each by one extremity to the extremities of the other two, so as to form a rigid framework; the rods make equal angles with each other, and the framework is suspended in a vertical plane by a pivot at its centre; if three weights be suspended from the outer extremities of the three rods find the position of equilibrium.

18. One end of a beam is connected with a fixed point by a hinge, about which the beam can revolve in a vertical plane; the other end is attached to a weight, by means of a string passing over a tack in the same vertical plane; find the position of equilibrium.

1.

MECHANICAL POWERS.

What force must be exerted to sustain a ton weight on a screw, the thread of which makes 158 turns in the course of 12 inches, and which is acted on by an arm 6 feet long?

2. In the annexed systems of pullies, compare P and W.

W

W

W

3. A weight W is sustained upon an inclined plane by a force P, acting by means of a wheel and axle, placed at the top, in such a manner that the string attached to the weight is parallel to the plane. Given R and r the radii of the wheel and axle, find the inclination of the plane.

4. What force is necessary to support a weight of 60 lbs. on a plane inclined at an angle of 60° to the horizon, the force acting horizontally?

5. Find the angle at which a given force must act, in order that it may just support a given weight on a given inclined plane.

6. When a given weight is sustained on a given inclined plane by a force in a given direction, find the pressure on the plane.

7. Given the weight, and the magnitude and direction of the sustaining force; find the angle of the plane.

8. Find the ratio of P to W in the various systems of pullies, taking into account the weights of the pullies.

9. Find the ratio of P to W in the single moveable pully, the strings making a given angle a with the horizon.

10. In the second system of pullies, if there be ten strings to the block, what power will support a weight of 1000 lbs. ?

11. In the third system of pullies, there being 6 pullies, what weight can be supported by a weight of 12 lbs. ?

12. What must be the length of a lever at the extremity of which a force of 1 lb. will support a weight of 1000 lbs. on a screw; the distance between two contiguous threads being 2 inch?

13. Two weights sustain each other on two inclined planes having a common altitude, by means of a string parallel to the planes; compare the pressures on the planes.

14. In the third system of pullies, there being 8 equal heavy pullies, find the ratio of the weight of one of the