SECTION XV. QUESTIONS IN PURE DYNAMICAL SCIENCE, NOT INCLUDING CENTRAL FORCES. 1821 1. Find the law of force by which a body may describe a rectangular hyperbola, the force acting in parallel lines perpendicular to one of its asymptotes. 2. Two pendulums, the lengths of which are L and l, begin to oscillate together, and are again coincident after n oscillations of the first pendulum. Given L to find l. 3. In what direction must a body be projected with a given velocity from a point in a given inclined plane, that the range may be the greatest possible? 4. Two bodies are projected from two given points in given directions and with given velocities; find their distance at the end of t". 5. A rod is placed in an inclined position, with one end upon a perfectly smooth horizontal plane; find the equation of the curve described by the other extremity whilst it falls. 6. Explain what is meant by accelerating force; and when P draws up Q in a machine, find the force accelerating Q's ascent, both when P and Q move with the same and with different velocities, the inertia and friction of the parts of the machine not being considered. 7. Two equal bodies A and B are connected by a string of given length: A is placed in a horizontal groove, and B hangs freely down, the string passing through an aperture which is continued along the bottom of the groove: a given velocity is given to A; find the position of B at the end of t". 8. A body descends by gravity, and describes in the nth second of its fall a space = p times the space described in the last but n; required the whole space. 9. Mention some of the experiments and observations, from which we may infer the truth of the second law of motion. 10. A given globe rolls down a given inclined plane in a medium resisting as the square of the velocity ; to find the time of describing a given space. 11. Two bodies A and B are placed upon a horizontal plane, and connected by a rigid rod without weight: a body C impinges upon a given point of the rod, in a given direction and with a given velocity ; define the motions of A and B, the body C not being connected with the system after the impact. 12. When a body is uniformly accelerated from rest, to find the space described in a given time. 13. Two equal weights are fixed, one at the middle point, and the other at the extremity of an inflexible and imponderable rod, which is suspended at the other extremity; if this compound pendulum be made to vibrate through small arcs, to find the time of its vibrations, 14. Apply D'Alembert's principle to find the velocity and time when p draws q over a fixed pulley. 15. A sphere of given radius is suspended by a point at a distance from its centre equal to its diameter. Find the time of its oscillation, and the point within the sphere at which it must be suspended so as to oscillate in the same time. 16. A body oscillates in a cycloid ; compare the whole tension of the string at any point with the weight of the body. 17. A body oscillates in a cycloid, in a medium the resistance of which « (vel.)? ; construct for the resistance at any point. 18. What must be the solid of revolution, so that when suspended by its vertex the centre of oscillation may be in its base ? 1822 19. Two equal heavy balls are suspended, by wires of the same given length, from the vertical axis of a machine, and are just in contact. How far will they separate from one another when a given angular velocity is communicated to the system? 20. A body falls down a given inclined plane, and, at the instant that it begins to fall, another is projected upwards from the bottom of the plane with a velocity equal to that acquired in falling through n times its length. Where will they meet? 21. A bucket descends into a well, unwinding a string from a cylinder of given weight and radius. What is the velocity acquired in falling through a given space, and the tiine of descent, the weight of the string being neglected ? 22. Three equal weights are placed at the angles of an equilateral triangle without weight, which is suspended by an axis perpendicular to its plane bisecting one of its sides; find the centre of oscillation. 23. A ball whose elasticity : perfect elasticity :: n : 1, is projected with a given velocity in a direction making an angle of 60° with the horizon, and when at its greatest height is reflected by a vertical plane ; determine where the ball will again strike the horizon and the whole time of flight. 24. Define the centre of spontaneous rotation ; shew generally how it may be found, and determine it when a straight rod of uniform density and given length is struck perpendicularly at a given point. 25. Find the whole times of ascent and descent of a body urged by the force of gravity in a medium whereof the resistance varies as the square of the velocity, and give Newton's constructions. 26. Two straight rods equal in length are suspended by their extremities, one being of uniform density, and the density of the other varying as the nth power of the distance from the point of suspension; and they make small oscillations in times which are as v5: 76. Required the value of n. 27. Compare the momentum of a paraboloid with that of its inscribed cone having the same base and vertex, when they both revolve round their common axis. 28. A string wrapped round a cylindrical annulus of uniform density whose radii are R and r, passes over a fixed pulley, and has a weight attached to it; find the space descended by the annulus in a given time. 29. The magnitudes of three perfectly elastic bodies are in harmonical progression ; prove that the momentum communicated to either of the extremes by the impact of the other equals the momentum of the mean moving with the velocity of the impinging body before impact. 30. If a body describe the arc of a cycloid by a force acting parallel to its base; prove that the force varies inversely as 2 sin 0 sin 20 ; 0 being the corresponding arc of the generating circle reckoned from the vertex. 31. Explain the different uses of a fly-wheel in machinery. 32. P descends vertically, drawing Q over a fixed pulley ; find the pressure upon the axis of the pulley, and its value when P is indefinitely increased. 33. A pendulum is composed of two thin wires of equal length, at right angles to each other at the point of suspension, and vibrating in their own plane. Find the time of a small oscillation ; and the angle at which they must be inclined to each other so that the time of oscillation may be doubled. 34. A weight W is raised upon a moveable pulley. The two extremities of the cord are wound in different directions about two cylinders, which have a common axis but different radii; and the power P descends unwinding a string from a wheel of given radius upon the same axis. What is the force which accelerates P's descent, when the strings are parallel to each other? 35. Explain Atwood's machine ; and mention some of the facts which it establishes in the theory of motion uniformly accelerated or retarded. 36. When the force by which a watch-balance is actuated varies as the nth power of the distance from the point of the spiral spring's quiescence; find the alteration in the daily rate, in consequence of a given change in the arc of vibration. 37. A second's pendulum of given length, in the form of a thin rectangular bar, suspended at the middle of its extremity by an axis perpendicular to its plane, is carried to the top of a mountain. The length of the bar is diminished by a given quantity in consequence of a change of temperature, the breadth remaining the same, and it loses t" in a day. What is the height of the mountain ? 38. A cannon ball weighing 24 lbs. strikes a wall with a velocity of 1700 feet. Find the weight of a beam, terminated by a hemisphere of the same diameter as that of the ball, which, when moved with a velocity of 10 feet, may penetrate to the same depth; and the weight of a similar beam, which may have the same effect in shaking the wall. 39. A body not affected by gravity falls down the axis of a thin cylindrical tube infinite in length, the particles of which attract with a force which varies inversely as the square of the distance. Find the velocity acquired in falling through a given space. 40. Shew that if a body oscillates in a cycloid, in a medium the resistance of which is constant, the successive altitudes to which it will rise are in arithmetical progression. 41. If a body is acted upon by any forces which would, if separately communicated, cause it to revolve about given axes with given angular velocities; find its axis of rotation, and its angular velocity; and apply the conclusion where there are three axes at right angles to each other. 42. Two bodies A and B descend from the same extremity of the vertical diameter of a circle, one down the diameter, and the other down the chord of 30°. Find the ratio of A to B when their centre of gravity moves along the chord of 120°. 43. A semi-circular area is placed with its vertex upon a horizontal plane ; find the time of one of its small oscillations. 44. A body projected in a direction parallel to the horizon, and acted upon by the force of gravity, describes a common cycloid; shew that the resistance of the medium, and the velo 0 0 city at any point, vary respectively as sin 2 O being 2 the corresponding arc of the generating circle. 1823 and cos |