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4. Two weights P and Q balance each other upon the surface of a sphere by a string of given length, passing over the highest point. Required the position of equilibrium.

5. Two weights sustain each other upon two inclined planes having a common altitude, by means of a string which is attached to each; find their position, taking into account the weight of the string, which is supposed to be uniform.

6. AD is horizontal, DC vertical, Q a weight connected with one extremity of a beam AB (moveable in a vertical plane about the point A) by a string passing over a pully at C in such a manner that CB is vertical. Find the relation between Q and the weight of the beam.

7. A weight W is sustained on an inclined plane by three forces, each equal to, one acting vertically upwards, another parallel to the plane, and the third parallel to the horizon; required the plane's inclination.

8. If three parallel forces acting at the angular points A, B, C of a triangle be respectively proportional to the opposite sides a, b, c; prove that the resultant may be supposed to act at the centre of the inscribed circle.

9. A uniform rod of given weight and length, has a weight attached to a certain point of it, and is placed with one end against a smooth vertical wall, the other upon a smooth horizontal plane; find the position of the weight, when a given horizontal force is just sufficient to prevent the rod from sliding when in a given position.

10. A uniform beam AB, of given length and weight, rests with one end on a given inclined plane, and the other attached to a string AFP passing over a pully at F given in position. Having given the weight Pfixed to the other end of the string, find the position in which the beam rests.

11. AC and BD are two smooth beams, of given weights, moveable in a vertical plane about the fixed points A and B in the same horizontal line; BD rests upon AC as a prop. Find the position of equilibrium.

12. A lamina, cut into the form of an equilateral triangle, is hung up against a smooth vertical wall by means of a string attached to the middle point of one side, so as to have a corner in contact with the wall; determine the position of equilibrium ; and shew that, if the string be beyond a certain length, equilibrium is impossible.

13. Two small smooth rings of equal weight slide on a fixed elliptical wire, the major axis of which is vertical, and are connected by a string passing over a snooth peg at the upper focus; prove that the rings will rest in whatever position they may be placed.

14. A pyramid, the base of which is a square, and the other faces equal isosceles triangles, is placed in the circumscribing spherical surface; prove that it will rest in any position, if the cosine of the vertical angle of each of the triangular faces be .

μ

15. A sphere of radius a is supported on a rough inclined plane, for which the coefficient of friction is μ, by a string of length, attached to it and to a point in the plane. Prove that the greatest possible elevation of the plane, in order that the sphere may rest when the string is a tangent, is 2 tan1 μ; and find the tension of the string and the pressure on the plane in the limiting position of equilibrium.

16. Perpendiculars are drawn from the angular points of a given triangle ABC upon the opposite sides, and another triangle is formed by joining the feet of these perpendiculars; prove that, if p, q, r be the distances of the centre of gravity of this triangle from the sides opposite to A, B, and C,

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DYNAMICS.

ELEMENTARY PRINCIPLES.

1. A POINT passes uniformly over a distance of 200 yards in the time 1. 6m: what is the numerical value of its velocity, according to the usual conventions respecting the units of space and time?

2. A point is observed to describe uniformly a feet in n seconds; supposing the unit of time to be 1 minute, what must be the unit of space in order that the numerical value of the point's velocity may be 1?

3. A man walks with a velocity represented by 2, and he finds that he walks 7 miles in 2 hours; if 1 foot be the unit of space, what is the unit of time?

4. A particle is moving with such a velocity, and in such a direction, that the resolved parts of its velocity in the directions of two lines perpendicular to each other are respectively 3 and 4; determine the direction and velocity of the particle's motion.

5. If 1 yard be taken as the unit of space, and 1 minute as the unit of time, what will be the numerical value of the acceleration of the earth's attraction?

6. A particle is proceeding uniformly along a certain straight line; suddenly it is observed to move off with unaltered velocity in a direction making an angle of 60° with the former direction of its motion; determine the direction and the acceleration of the impulse to which this change of the motion is due.

7. If a be the distance between two moving points at any time, V their relative velocity, u, v, the resolved parts of V in and perpendicular to the direction of a, shew that their distance

av

when nearest to each other is, and the time of arriving at this

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8. If a shilling be the unit of money, £1000 a year the unit of income, and an inch per minute the unit of velocity, find the unit of space.

9. Two vessels are sailing with velocities V and V'in directions which make an angle a with each other; if be the angle between the line joining them, and the direction in which a gun must be fired from one of them in order to strike the other, find for any given position of the vessels, and shew that its greatest value is given by the equation

v sin 0 = ( V2 + V22 – 2 VV' cos a)3,

where v is the velocity of the shot, supposed uniform.

FALLING BODIES.

1. Find the depth of a well by dropping a stone into it, the velocity of sound being supposed to be known.

2. A particle is projected upwards with a velocity which will carry it to a height 2g feet; after how long a time will it be descending with a velocity g?

3. Find the velocity with which a particle must be projected upwards from the foot of a tower, so as to meet halfway another particle let fall at the same time from the top of the tower.

4. A balloon is ascending vertically with a given velocity, and a particle is let fall from it, which reaches the ground in t": find the height of the balloon at the moment of the body leaving it.

5. A particle is observed to fall the last a feet of its descent from rest in t": find the height from which it fell.

6. A particle has fallen through a distance of half a mile; what was the space described in the last second?

7. The space described in the fifth second of its fall was to the space described in the last second but four as 1 : 6; what was the whole space described by the particle?

8. A particle is projected upwards with a velocity of 64 feet in 1"; how far will it ascend before it begins to return?

9. With what velocity must a stone be projected from the top of a tower, 250 yards above the sea, that it may reach the water in 6"?

10. A particle falls through a distance a feet at two different places on the earth's surface; and it is observed that the time of falling is t" less, and the velocity acquired m feet greater at one place than at the other: compare the acceleration of gravity at the two places.

11. A stone dropped from a bridge strikes the water in 21"; what is the height of the bridge? Also, if the stone be projected downwards with a velocity of 3 feet per second, in what time will it strike the water?

12. Two balls are projected at the same instant towards each other, from the two extremities of a vertical line, each with the velocity which would be acquired in falling down it. Where will they meet?

13. A falling particle is observed to describe in the nth second of its fall a space equal to p times that described in the - 1th required the whole space described.

N.

14. A particle is projected vertically upwards, and the time between its leaving a given point and returning to it again is given; find the velocity of projection and the whole time of motion.

15. If a, be the measure of a uniform acceleration when ₺

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