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PRELIMINARY HONOUR DYNAMICS.

1. The general hydrokinetic equations (Euler's equations) using the notation of the lectures, are:

Apply these to the case of a vessel of liquid rotating with constant angular velocity about a vertical axis, under the force of gravity alone, to show that the free surface is a paraboloid of revolution.

2. Define -Stress, strain, moment of inertia, centre of percussion, elastic limit, coefficient of friction, angular momentum, vector product.

3. (a) A weight of 5 lbs. is supported by a spring. The stiffness of the spring is such that putting on or taking off a weight of 1 lb. produces a downward or upward motion of 0.04 ft. What is the time of oscillation, neglecting the mass of the spring.

(b) An iron hoop hangs on a nail. Find its period of oscillation as it vibrates under the force of gravity in its own plane.

4. A gate swinging on two hinges is capable of rotation about a fixed axis. A blow is struck causing the gate to swing. Write 3 equations of translation and 3 cquations of rotation and reduce them to express the angular velocity and the reaction at the hinge bearings, in terms of the components of the blow.

5. A ladder resting on the ground leans against a wall. The coefficient of friction for each end of the ladder and the distribution of load on the ladder being given, find the condition under which the foot of the ladder will not slip.

6. Three laborers are driving a bolt into the ground, and each strikes the bolt 16 times a minute with a sledge hammer weighing 12 lbs. For each stroke the head of the hammer is raised 5% feet above the bolt, and brought down with a uniform pressure in the time it would have taken to fall freely. Express the rate at which the three work together as a fraction of a horse power.

7. (a) Find the centre of gravity of a hemisphere of homogenious material in terms of its radius.

(b) Find the radius of gyration of a 60° sector of a uniform circular plate, if the mass of the whole plate is M.

FINAL HONOUR DYNAMICS II.

1. (a) Explain the term Vector Cross. To what may it

be reduced?

(b) What is meant by the central axis of a system of forces, and how may its equations be determined?

2. Show that a rigid body acted upon by a system of forces starts to move through space as though all mass were concentrated at the C. of G., and that it rotates at the same time about the C. of G. as though it were fixed.

3. Show that the values of the moment of inertia of a rigid body for different axes are not independent, and how if the value is known for one axis, it may be found for any other.

4. A rigid body has two points in it fixed. Show how to determine the reactions at these points when the body is acted upon by any system of forces.

5. Show how to locate the centre of percussion for the body of question 4.