Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση
[blocks in formation]

and find its value, having given that y(x) has P(x) for its differential coefficient.

12. Find a formula for the volume of a solid of

revolution.

The cur

yt = c(a - x)(x - 1) revolves about the axis of x; find the volume generated.

MIXED MATHEMATICS.-Part I.

The Board of Examiners.

1. Shew how to find the magnitude and direction of

the uniform acceleration of a particle moving in a plane, when the magnitude and direction of the velocity at two instants are known.

A body of mass 1 kilogramme is moving North with a velocity of 5 kilometres per hour at one instant, and an hour afterwards is moving East with a velocity of 6 kilometres per hour. Find the magnitude and direction of the force on it in dynes.

2. Shew how the ratio of the masses of two bodies

may be found by allowing them to impinge on one another, and observing the alterations in their velocities.

A sphere of mass m overtakes and impinges directly on a sphere of mass m', the coefficient of impact being e. Shew that the former sphere cannot have the direction of its motion reversed

by the impact unless m < em'. 3. Prove that the acceleration of a particle moving

with uniform velocity v in a circle of radins r is

[ocr errors]

Shew that a simple conical pendulum of length 1 metre inust make one revolution in 1.7 seconds nearly if it makes an angle of 45° with the vertical.

4. Find the time of a small oscillation of a simple

pendulum, and shew that a small percentage of increase in the length of the pendulum, or of decrease in gravity, increases the time of oscillation by half

that percentage nearly. 5. Find the resultant of two parallel forces on a rigid

body.

A circular cylindrical cup with a flat bottom is made of thin smooth sheet metal of weight w per unit area. It is placed on a horizontal table, and a uniform rod of weight W is placed in it. Shew that the rod will upset the сир

if its length is greater than 4m2 + h

2 + (ar2 + 2arh

πη2

W where h is the height of the cup and r its radius.

n

[:

X

[ocr errors]

r

6. Prove that it is necessary and sufficient for the

equilibrium of a rigid body acted on by forces in one plane, that the sums of the moments of the forces about three non-collinear points should vanish.

A uniform heavy rod rests with one end on a smooth horizontal plane, and the other against a smooth vertical wall. The lower end of the rod is connected with a point of the intersection of plane and wall by an elastic string of natural length V2j3 times that of the rod, and of modulus equal to the weight of the rod. Shew that there is equilibrium when the rod makes an angle of 45° with the vertical.

[ocr errors]

7. Two equal uniform rods AB, AC, each of weight

w, are smoothly jointed to one another at A, and to fixed points B, C, such that BC is vertical and equal to AB. A weight W being hung at A, find the reactions at the joints.

a

[ocr errors]

8. Shew that, neglecting friction, the work done by a

machine, working steadily, is equal to the work done on it.

Apply this principle to find the mechanical advantages of a smooth screw, and of that systein of pulleys in which the same string passes over

all the pulleys. 9. Define the intensity of fluid pressure at a point.

The safety-valve of a steam boiler is a circle of radius r, whose centre is at a distance a from the fulcrum, and the weight W on the valve is at a distance c from the fulcrum. Find the steam pressure when the valve is about to open.

а

a

10. Prove that the centre of pressure of a triangular

area immersed in heavy liquid, with one side in the surface, is one-half the way down the median through the lowest point.

If the triangle is of height h, and its plane is vertical, shew that the addition of a depth k of liquid will raise the c.p. a height hk|(6k + 2h).

a

11. Investiyate the pressure at any point of a mass of

water rotating steadily as if rigid about a vertical axis.

If a particle of density •9 revolves with the water making 50 revolutions a second, and is at a distance of a decimetre from the axis, sbew that there is a radial force on it about 113 times

its weight. 12. Investigate an expression for the density of the

air in an air-pump after n strokes, the clearance and leakage being neglected.

MIXED MATHEMATICS.-Part II.

The Board of Examiners. 1. Shew that the tension in a circular hoop rotating

with angular velocity w is mw2p2 where m is the mass per unit length.

If the hoop is of iron (sp. g. 7-8) and breaks with a tension of 20 tons per square inch, shew that its velocity must not be greater than 650 feet per

second.

2. Shew that if a particle describes an ellipse under a

central force in a focus, the force varies as the
inverse
square

of the distance,
Shew from the equation of energy and by
normal resolution that a particle can describe an
ellipse under two Newtonian central forces, one in
each focus; the square of the velocity being the
sum of the squares of the velocities at the same
point when each force acts alone.

[ocr errors]

ñ are

3. Prove that if a plane curve rolls, without slipping,

on a fixed curve, the angular velocity w and the
velocity v of the point of contact are connected
by the equation w=v (v + k') where
the curvatures of the curves at the point of
contact.

A circle of radius a rolls on one of radius b, making n revolutions a second. Find the velocity of the point of the rolling circle furthest from the fixed circle.

4. Two heavy particles move in smooth vertical

circles of different radii. Find how the particles must be started in order that their velocities may hear a constant ratio to each other when they are at equal angular distances from their lowest points, and shew that the reactions in corresponding positions will then be as the masses.

6. Investigate a graphic construction for the resultant

of a system of forces in one plane, and obtain the graphical conditions of equilibrium.

6. Prove Pappus's theorem that the volume of the

solid of revolution generated by a plane area

« ΠροηγούμενηΣυνέχεια »