7. Explain what is meant by the virtual velocity and virtual moment of a pressure acting on a point, and state the principle of virtual velocities.

If X and Y be the rectangular components of a pressure P, shew that the virtual moment of P is equal to the sum of the virtual moments of X and Y. When will this equality of virtual moments express the principle of virtual velocities?

Parkinson's Mechanics, Arts., 116, 117.

Let 0 (fig. 16), the point of application of the force P, be supposed displaced to O'. From O' draw O'x, O'y, O'p perpendiculars to OX, OY, and OP.

Since P is the resultant of X and Y, therefore the resolved part of P in any direction is equal to the sum of the resolved parts of X and Y in the same direction; therefore P cos O'OP= X cosx00'+ Y cos O' Oy; therefore

P.00' cos O' OP=X.00' cosx00'+ Y.00' cos O' OY, P.Op=X.Ox+ Y.Oy,

or

that is virtual moment of P= sum of virtual moments of X and Y.

If we change the direction of the force P, it, together with X and Y, would form a system in equilibrium, and the above equation with the sign of the left-hand term changed would express the principle of virtual velocities.

8. Define a couple, and shew that any system of forces acting in one plane may be reduced to a single force and a couple; hence, examine the conditions of equilibrium of the forces.

Todbunter's Mechanics, Arts. 65, 84, 86, 87.

9. A uniform beam of given weight rests upon a prop with one end against a smooth vertical wall and a weight is suspended from the other end of the beam; find the position of equilibrium.

1

Let AB (fig. 17) be the rod, 2a its length, C the peg, c its distance from the wall, W, the weight of the rod, W, weight attached to the end B, the inclination of the rod to the horizon when it is in the position of equilibrium.

2

1

Resolving vertically we have R, cos 0 = W1+ W1. Taking moments round A,

R1x AC= Wa cose + W2a cose;

therefore we have

( W1 + W2) c sec20 = ( W2+2 W2) a cose,

1

1

10. When friction exists between two substances, explain what is meant by the limiting angle of resistance. If the limiting angle of resistance be 30°, what is the coefficient of friction?

Describe the construction of the screw considered as a mechanical power, and find either of the limiting conditions of equilibrium on the screw when friction is taken into account.

Parkinson's Mechanics, Arts. 57, 111, 112, 113.

11. When is equilibrium said to be stable, unstable, and neutral? Give examples of each kind of equilibrium. If a body rest with a convex spherical surface on the interior of a concave spherical surface, examine the mathematical condition of stability.

Parkinson's Mechanics, Arts. 81, 83, 84.

A sphere of radius rests on a concave sphere, radius R; if the sphere is loaded so that the height of its centre of gravity from the point of contact is r, find R in order that the equilibrium may be neutral.

Result, R=4r.

12. State Guldinus's properties, and prove the property for finding either the surface or the volume of a solid formed by the revolution of a plane curve or curvilinear area.

Determine either the volume or the surface of a ring made by the revolution of a regular hexagon; the radius of the circle that would circumscribe the hexagon is a, and b is the distance of the centre of the hexagon from the axis about which it revolves.

Perimeter of hexagon = 6a, distance of its C of G from the axis of revolution = b; therefore surface of ring = 2πb × 6α = 12πab.

Similarly area of the hexagon

=

3a2 √(3);

2

PURE MATHEMATICS (3).

WEDNESDAY, 13TH DECEMBER, 1876. 2 P.M. TO 5P.M.

1. Prove that an equation of even degree which has the last term of contrary sign to the first has at least two real roots of contrary signs.

Todhunter, Theory of Equations, Art. 21.

2. Transform the equation x3-4x-3=0 into one whose roots exceed by the corresponding roots of the given equation.

3. Prove that in any equation, complete or incomplete, the number of positive roots cannot exceed the number of changes in the signs of the coefficients. Todhunter, Theory of Equations, Art. 63.

4. It being given that the limit of (1+x) when x is indefinitely increased is e, show that the limit of (1+ax) is eo.

1

5. Shew that the differential coefficient with respect to x of the product of u and v, two functions of is

dv du
и +v
dx dx

Todhunter, Diff. Cal., Art. 29.

Differentiate the following functions with respect

√(1 + sin x) + √(1- sin x)

=

2

2

V(1 – sinæ) * V(1+sinx) V(1-sinx)

6. If u be any function of (at + x), prove that

7. Prove that if f(x) has a maximum or minimum value when xa, then f' (a) = 0.

Todhunter, Diff. Cal., Art. 211.

Find a value of x for which " (1-x)" is a maximum, m and n being positive.

Now for a maximum or minimum value of y this = 0.

Hence we obtain x=0 or x=

of x makes y a maximum.

m

which latter value

m + n