Find the resultant of any number of forces acting on a rigid body in a plane at one given point.

Three equal forces act from the angles of a triangle to the centre of the circumscribing circle; prove that the resultant passes through the centre of gravity of the triangle.

Define the moment of a force, and prove that the moment about any point of the resultant of any forces in one plane is equal to the algebraic sum of the moments of the forces about the same point.

Define the centre of gravity of a body, and prove that the work done by gravity on a system of bodies in motion is the same as if the whole system were condensed into the centre of gravity.

Four uniform heavy rods a, a, 2a, 2a are formed into a reentrant polygon, the reentrant angle being formed by the two smaller rods. If the figure balance about this angular point in any position, find the angle.

State the usual laws of friction, and explain whether they are all independent; find the force required to pull a body up a given rough inclined plane.

A rough hoop in the form of a parabola is fixed with its plane page 2 vertical and axis inclined at an angle a to the vertical; and a heavy rod rests inside it in contact with the hoop, the coefficient of friction being tan Λ. If in the limiting position of equilibrium, the rod passes through the focus, prove that a=2Λ.

Investigate the theory of Atwood's machine.

A weight P descends vertically drawing Q up an inclined plane by a string which passes over a pulley at the top of the plane; determine the motion. What part of P's weight must be taken from it, without interfering with its velocity, when it has pulled Q half up the plane, that Q may just reach the top of the plane?

Find the range of a projectile on an inclined plane passing through the point of projection.

Two particles are projected from the same point of a plane of inclination i, with equal velocity u, and in directions perpendicular to each other, so as to have equal times of flight. Show that the difference of their ranges bears to their sum a ratio = tan i.

Distinguish the whole pressure from the resultant pressure, and explain how each is to be found.

A weightless cylinder is filled with water and hung up by a string attached to a point on the rim of one of the ends; compare the whole pressure and resultant pressure on the curved surface.

Show how to determine the conditions of equilibrium of a body at rest in a liquid and supported in any manner.

A long cone, of density ⍴, whose thickness is everywhere inconsiderable compared to its length, and which is loaded at the thick end, rests with this end on the bottom of a pond and the other end sticking out of the water; prove that the fraction x of the rod uncovered is given by 3x⁴−4x³+1 =⍴.

Explain how Boyle's Law for gases may be verified experimentally. What connection is there between it and Dalton's Law?