IV. STATICS AND DYNAMICS.

I. Show how to place three forces, represented by the numbers 3, 4, 5, at a point so as to balance each other.

Would it be possible for the forces 2, 3, 5 to make equilibrium ?

2. Three equal forces, P, diverge from a point, the middle one being inclined at an angle of 60° to each of the others. Find the resultant of the three.

3. A uniform lever, weighing 30 lbs. and 16 feet long, is loaded at either end with weights of 20 and 30 lbs. respectively. Find where the fulcrum must be placed for equilibrium.

4.

Prove that the moments of any two non-parallel forces, about any point on their resultant, are equal and opposite.

If two forces form a couple, show that the algebraical sum of their moments about any point in the plane is constant.

5. Show how a weight of 2 lbs. can be balanced by a weight of 1 lb., by means of pulleys; and show that the former will rise through half the space which the latter descends.

6. State, and prove, what power acting along a smooth inclined plane is required to support a weight resting on the plane.

A heavy string is placed with part of it resting on a given inclined plane, the remaining part hanging vertically over a small pulley at the top of the plane. Find the point of the string which must be placed over the pulley, for equilibrium.

7. Explain, and prove, the Parallelogram of Velocities.

Decompose a given velocity, v, into two components whose directions are each inclined at 30° to its own.

8. What is meant in Dynamics by the Mass of a body? What result of observation proves that the Weights of all bodies are proportional to their Masses?

9. Explain clearly what is meant by saying that g=32, where g is the acceleration due to gravity.

If the unit of time were altered to 2 sec., what would be the value of g; and why?

10. In what time will a body fall from rest through 100 ft.? If it be retarded in its fall by the tension of a string attached to it, so as to occupy 5 sec. in the fall, what is the tension of the string, the weight being given?

II.

A body falls freely through 100 ft. from rest, with what velocity will it reach the ground?

If, instead of falling from rest, it be projected downwards so as to reach the ground with twice the former velocity, find the initial velocity.

12. A ball is rolled across a smooth table, and leaves it with a given horizontal velocity. What curve does it describe during its fall?

If v be the initial velocity, and 1⁄2 the height of the table, find the horizontal distance from the table of the point where the ball reaches the floor.

FURTHER EXAMINATION.

V. PURE MATHEMATICS (1).

[Full marks may be obtained for about four-fifths of this paper.]

[Great importance is attached to accuracy.]

I. Show that, if from the focus of a parabola a perpendicular be drawn to any tangent, the point of intersection lies on the tangent at the vertex.

Having given the extremities of a focal chord of a parabola, and also the focus, find the position of the vertex.

2. Prove that the area of the parallelogram of which two semidiameters of an ellipse are adjacent sides is invariable, if the diameters be conjugate.

Show that the sum of the lengths of two conjugate diameters of an ellipse is never less than the sum of the lengths of the principal axes.

3. Define a hyperbola by means of the relation between the distances of any point upon it from a given point and a given line, and deduce from the definition the property that the difference between the focal distances of a point on the curve is invariable.

Two circles in the same plane are external to each other. Show that the locus of the centre of a circle touching them externally is a hyperbola, and determine the position of the asymptotes.

4. Prove that the focal radii of a point of a central conic section make equal angles with the tangent at that point.

Show that an ellipse and a hyperbola which have the same foci cut one another at right angles.

5. Establish the focus and directrix property of the section by a plane of a right circular cone.

Determine the eccentricity (1) of a circle; (2) of a pair of parallel straight lines.

6. Find the rectangular coordinates of a point dividing in a given ratio the straight line joining two points with given coordinates.

Examine the case when the given ratio = - I.

7. The equations to two straight lines being given, find the equation to a straight line joining a given point to the point of intersection of the straight lines.

Find the equation to the straight line passing through the origin and the intersection of the straight lines given by

8.

Show how to determine the points where the straight line represented by

cuts the locus of the equation

y=mx+b

9. Find the equation to the normal to the parabola given by y2=4ax at the point whose coordinates are (x', y'). Show that not more than three normals can be drawn to a parabola from a point in its plane.

10. Show how to determine the pair of diameters common to the circle given by x2+y2=k2 and the locus of ax2+ by2+zhxy+c=o. Under what conditions will the diameters become coincident?

II.

given by

Determine the locus of the intersection of the straight lines

What connection is there between the angle 2 tan-1t and the point on the locus where the two straight lines intersect?

12.

Discuss and trace the loci of the following equations:—

(3) x=a cos (0+a), y=b cos (0 +B), where 0 is a variable parameter.

VI. PURE MATHEMATICS (2).

[Full marks may be obtained for about four-fifths of this paper.]

are and I, corresponding to the values 1 and 2 of x.

I

4.

Prove that, if A varies as Bm when C is constant, and if A varies as C when B is constant, then will A vary as Bm Cn when both B and C vary. Determine the resistance of the air to a projectile 16 inches in