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University of New Zealand.

B.A. Examination for Honours, 1885.

Mathematics.

Paper d. Mechanics.

Examiner: Prof. C. Niven, F.R.S.

1. State the Second Law of Motion, and explain its relation to the parallelogram of forces, and to the definition of the unit of force.

Three equal spheres of radius a are suspended from the same point by strings of equal length b, and hang in a vertical plane: prove that the tensions of the two outside strings are each equal to the weight (W) of a sphere, and that the pressure between each two spheres: W:: 2 a: a + b.

2. Find the conditions of equilibrium of a rigid body under a system of forces in one plane, and show that they will be satisfied if the sum of the moments about 3 points not in the same straight line separately vanish.

Show that any force in the plane of a given triangle may be resolved into 3 forces acting along the sides of the triangle; hence show that if L₁ L₂, L₃ be the sums of the moments of a system of forces in one plane about three points A, B, C, in the plane of the forces, the sum of the moments about any fourth point in the plane is are the perpendiculars from the fourth point on the sides of the triangle ABC, a,b,c its sides, and Δ its area.

3. Explain generally how you would deal with the equilibrium of a statical system consisting of a number of jointed rods.

A rhombus is formed of four weightless rods jointed freely at the angles, the opposite angles being joined by elastic strings cut from the same uniform piece. Assuming Hooke's Law page 2 that the extension of a string varies as the stretching force, prove that if Λ, Λ′ be the lengths of these strings in the position of equilibrium, l,l′ their unstretched lengths, then

4. Find the equation of the curve in which a heavy uniform string, attached to two fixed points, will hang.

A and B are two points in the same horizontal plane, and a string attached to A passes over a small pulley at B and hangs vertically, the weight of the vertical part B C being sufficient to keep the other part A B nearly straight: prove that if D be the distance of the middle point of the string below the straight line A B, 8 D · BC = AB², nearly.

5. Find the mechanical efficiency of a screw, on the supposition that the threads are smooth.

Why is it impossible to withdraw a screw nail without unscrewing it?

6. Find the law of force under which a body will describe a parabola, the force being constantly perpendicular to the axis of the curve.

A particle moves uniformly in an ellipse, find the components of the force which act upon it at every instant.

7. A body moves in a plane curve, find the acceleration along the radius vector.

A rod O A, movable in a vertical plane about O, revolves with uniform angular velocity ω from the vertical to the horizontal position, while a smooth ring under the influence of gravity moves from rest at A towards O: show how to find the motion of the ring.

8. Obtain the equation of a central orbit

If the angular velocity of a particle describing an orbit about a centre of force be inversely proportional to the velocity, show that the equation to the path is r³ sin 3θ=a³.

9. A body moves, under gravity, on the arc of a smooth vertical curve, find the velocity at any point of the motion and the pressure on the curve.

A particle slides from the top of a smooth vertical circular hoop, find where it will leave the curve.