Analytical dynamics, being a synopsis of leading topics in the analytical theory of dynamics with numerous examples and selections from Newton's Principia and other sources, by Arthur S. Hathaway.

end so as to be inclined at a fixed angle to the vertical; to find its
angular velocity, and the direction of the pressure at the fixed
point.
42. A hemisphere oscillates about a horizontal axis which
coincides with a diameter of the base. Its base is inclined at an
angle of 60 degrees to the horizon at the beginning of the motion..
To find the pressure on the axis at any time, and the maximum
pressure. Also the center of oscillation.
43. A homogeneous sphere is suspended by a fine wire attached to a fixed point on its surface. It is turned through n
revolutions and released. Determine the untwieting motion, -lhe.
elasticity of torsion being proportional to the angle.
44. If a partcle be attached by a string of glvnri length to a
po(ilt in a rough horizontal plane, and a given velocity be collmlunicated to it, perpendicular to the string supposed tight, Jind the
lension of the string at any time, the time in whlich it will be
brought to rest, the whole arc described.
45. Two equal rings, P, Q, slide on a;tri nr which passes
round fixed pegs A, B, in a smooth horizontal plane. The rings
are brought together, and projected with equal velocities so as to
kl(:ep he string stretched symmletrically. Show that the path of'
(ec.h is a paraboil, nIld that the ten-sion of the stri.ng vaTtries inveriv as the liiauee of eaci from the nearer I; —
46. A particle moves in an elliptic tube, under the action of
a rod with its ends in the focil, each element of which attracts inversely as the square of the distance. Show that its speed is constant, and find the pressure of the tube at the end of the minor
axis.
47. Conservation of Areas. If a system of particles be subject only to mutual actions, and the products be made of the mass
of every particle into the projection on a given plane of the area
described by its radius vector, show that the sum of those products.
varies uniformly with the time.
48. A uniform straight plank rests with its middle point on
a rough horizontal cylinder, their directions being perpendicular
to each other. The plank is slightly displaced so as to remain in
contact with the cylinder without slipping; to determine its period
of vibration.
49. An inelastic beam, one end fixed to a horizontal axis,.
falls from a given position and impinges against an immovable
obstacle; to determine the shocks at the obstacle, and on the axis.
Also, determine the position of the obstacle in order that there may
be no shock on the axis.
NOTE:-Let R, R' be the impulsive reactions at the two places

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