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.

-vertical, OL, and I, the moment of inertia about the axis. Computing moment of the force (gravity) about the axis,
I.D26 ---mga sinO.
Compare this with the equation for the simple pendulum of length
1, at the same angle, at any time t, 1.D2= —gsin0, and we find that
l=I/ma, which is called the length of the equivalent simple pendulum.
Produce OG to C, making OC==I/ma; C is called the center
of oscillation as to the center of suspension O. Let k be the radius,of gyration about a parallel axis through the center of gravity G;
then I=m (a2+ c2). Thus OC=a+-b, where
b —GC=k2/a.
Conversely, if C be made center of suspension with a parallel
axis, its center of oscillation will be on CG produced, at a point
which is a distance k2/b, or a beyond G, which makes it the point
0. Thus, centers of suspension and oscillation are interchangeable.
EXAMPLES VI
1. Prove that the whole work done in raising a system of
bodies through different heights is the same as raising the entire
weight through a height equal to that which their center of gravity
is raised.
2. Find the work performed in moving a ton 100 yards on a
uniformly rough horizontal road, the coefficient of friction being
*one-tenth.
3. Show that the same work is expended in drawing a body
up an inclined plane, subject to friction, as would be expended by
drawing it along the base and raising it vertically upward.
4. If two particles attract each other directly as the product
of their masses, and inversely as the square of the distance between them, find the work done, whet they have moved from an
infinite distance apart to the distance r.
5. Find the potential outside, upon, and within, a spherical
shell, of uniform mass per unit area. Derive thence its attraction
upon a particle in the same situations.
6. Show hence, that a solid spherical mass, whose density at
any point varies only with the distance from the center, attracts
an outside mass as if its whole mass were concentrated at the
center.
7. Determine the law of force wthin a tube along a diameter
of the earth.
8. A flat pivot presses against a rough plane; find the work
done against friction in one revolution of the pivot.
9. A ball moving with a velocity of 1,000 feet per second has

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