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.

Note. The normal acceleration of gravity at (x,y) is gdx/ds
inwards, and tan = —dy/dx. Prove the tube exerts the inward
normal acceleration Vag (h-a) /V (a-y) 3, at the point (x,y),
etc.
45. A heavy particle hangs freely from a fixed point by a
fine elastic wire. The wire produces an acceleration upward proportional to its extension of length and the extension for which the
particle hangs at rest is I units. If the particle be pulled down a
units, and released, show that it describes harmonic motion of
amplitude a, and that its period is the same as a simple pendulum
of length 1 units.
46. A horizontal platform vibrates up and down harmonically
in a period of one second. Find its greatest amplitude in centimeters such that particles may rest upon it undisturbed.
CENTRAL FIELDS
47. A partical starts from rest at a given point
in a central field whose acceleration towards the center is
proportional to the distance from the center. Show that its path is
a straight line through the center, and that it executes harmonic
vibration about the center whose period is independent of the starting position, and inversely as the square root of the absolute acceleration of the field.
48. If a small hole were cut through the earth along a
diameter we should have the field in the previous example. Find
the time for a particle dropped in at one end to emerge at the other,
the diameter being 8,000 miles. Also the velocity with which it
reaches the center.
49. A particle is placed on the line joining two centers, the
acceleration towards each being proportional to the distance of the
particle. Find the center of the motion, and the time of vibration,
in terms of the absolute accelerations.
50. If a particle be acted on by repulsive acceleration from
a fixed center, proportional to its distance from the center,:.nd
start with velocity in a line with the center, show that its motion
is in a straight line, and hyperbolic.
51. The earth produces an acceleration towards its center
which varies inversely as the square of the distance from the center.
A particle is projected vertically from the surface with speed T.
To what height will it rise? Compare with the height if the acceleration were constant. Find the least value of V such that the
particle never returns to the earth.
52. A particle moves on a given curve in the earth's field.
Its arc distance from a fixed point being s, and its distance from
the center being r, at time t, show that the tangential acceleration

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