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m

must be

and the velocity with which it will move

e

after impact is ev, if v be the velocity of the first ball

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after impact = e3v, &c.; therefore the mass of the nth

ball =

m

and the velocity with which it moves off

after impact e-1v.

=

9. Given the accelerations of a particle in two directions at right angles to another, find the accelerations along the tangent and normal to its path.

If a particle attached to a string is made to revolve in a vertical circle, and if the tension of the string at the highest point be zero, what will be the tension at the lowest point?

Parkinson's Mechanics, Arts. 105, 109.

10. Prove that the time in which a heavy particle will slide down any arc of an inverted cycloid to the lowest point is the same.

Parkinson's Mechanics, Art. 102.

If equal the length of an imperfectly adjusted seconds pendulum which gains n.seconds in an hour, and equal length of one which loses n seconds at the same place, find the true length of the seconds pendulum at that place.

Let be the true length of the seconds pendulum. Also let N=60 equal number of seconds in one hour.

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11. If a pendulum which beats seconds at the surface of the earth supposed spherical with a radius of 4000 miles loses 10 seconds in 24 hours at the top of a mountain, find the height of the mountain and the number of seconds which the same pendulum would lose, in 24 hours half way up.

Parkinson's Mechanics, Art. 104.

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half way up the mountain the pendulum would lose 5 seconds in 24 hours.

STATICS.

WEDNESDAY, 13TH DECEMBER, 1876. 10 A.M. TO 1 P.M.

1. By what assumptions does Geometry become applicable to establish the fundamental propositions of mechanics? Pressure, tension, reaction, friction are included under the term force; give examples of the

mode of action of each, and state how they are measured statically.

Parkinson's Mechanics, Arts. 4, 10, 41, 52.

A weight at rest is placed on a smooth table; what forces act upon it? If the table be inclined, shew that some other force must be introduced to keep it at rest. If three equal forces act on a particle, how must they be arranged so as to keep it at rest.

Parkinson's Mechanics, Arts. 5, 23.

2. State the proposition known as the parallelogram of forces, and shew how it may be proved by experi

ment.

Parkinson's Mechanics, Art. 18.

chanics, Art. 28.

Todhunter's Me

A stream is 96 feet wide; a boat is dragged down the middle of the stream by two men on the opposite banks pulling by ropes each with a pressure of 100 lbs. ; the ropes are attached to the same point in the boat, and the length of each rope from the boat to the bank is 60 feet, find the resultant pressure on the boat down the stream.

Resultant pressure on the boat = 120 lbs.

3. State and prove the condition of equilibrium of a straight lever without weight acted on by any two forces at its extremities.

Parkinson's Mechanics, Art. 91.

AB a uniform rod of weight W, movable in a vertical plane about a hinge at A is sustained in equilibrium by a weight P attached to a string BCP passing over a

smooth peg C, AC being vertical, if cos ACB= shew that AC is = AB.

P

W'

Let ACB=0 (fig. 15), ABC=0', and let the length of the rod be 2a.

Taking moments about A, we have

Wa sin (0+0') = 2Pa sin0'; but P= W cos 0, therefore sin cose' + cose sine' = 2 sin e' cose,

or

therefore

cot 0 = cot 0';

0'0, and AC=AB.

=

4. Define the centre of gravity of a solid or system of particles. Shew that every solid or system of particles must have a centre of gravity, and cannot have more than one such centre.

Assuming the position of the centre of gravity of a pyramid with a triangular base, shew that its position is not altered by placing equal weights in each of the four solid angles of the pyramid.

Parkinson's Mechanics, Arts. 70, 79.

5. How is the work done by an agent estimated? What is the numerical measure of a horse-power? When weights are raised through different heights, prove that the whole work expended is equal to the work that would be expended in raising a weight equal to their sum through the same height as that through which the centre of gravity of the weights has been raised.

Todhunter's Mechanics, Arts. 192, 198.

In how many hours would an engine of 18 horsepower empty a vertical shaft full of water, if the diameter

G

of the shaft be 9 feet, and its depth 420 feet, the weight of a cubic foot of water being 62.5 lbs.

The weight of water raised =π.19' x 420 × 62.5 lbs., and the height through which the C of G of the whole has been raised = 210 feet.

Therefore the number of minutes required to empty the shaft

=

T.81.420.62.5.210

4 x 18 x 33.000

= 9 h. 50 m. 23.32 s.

6. What is meant by the mechanical advantage of a machine? Is any mechanical advantage obtained in raising a weight by means of a fixed pulley? Investigate the mechanical advantage when a weight is raised by means of a smooth moveable pulley, the parts of the string on each side of the pulley not being parallel. At what angle of inclination will the mechanical advantage cease? If the directions of the string on the moveable pulley are at right angles to each other, what is the greatest weight that can be raised by a man standing on the ground whose own weight is 12 stone, when he pulls at the free end of the string which passes over a fixed pulley having its centre on the same horizontal line as the point to which the fixed end of the string is attached?

Parkinson's Mechanics, Art. 103.

In the case of a single moveable pulley the mechanical advantage ceases when the angle between the two parts of the string is > 120°.

The greatest weight that can be raised in the case of the strings being at right angles

= 2 × 12 cos 45° = 12 √(2) stones = 158.37 lbs.

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