Part 3

The Center of Gravity

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Table of Contents

Finally, if the center of gravity moves along a line given with respect to the moving plane; by taking the axis of ζ for this line, we have dηdt=0\frac{d\eta}{dt} = 0 , dξdt=0\frac{d\xi}{dt} = 0 , and consequently,

Md2ξdt2=∑X−∑Xe. M \frac{d^2\xi}{dt^2} = \sum X - \sum X_e.

If we now wish to examine what becomes of the equations of the areas, we must, in equation (B), take virtual velocities of rotation about one of the coordinate axes; for example, set

δz=0,xδx+yδy=0, \delta z = 0, \quad x \delta x + y \delta y = 0,

or

δz=0,δxy=−δyx. \delta z = 0, \quad \frac{\delta x}{y} = -\frac{\delta y}{x}.

We shall thus obtain

$$ \sum m \left( y \frac{d^2x}{dt^2} - x \frac{d^2y}{dt^2} \right) - 2r \sum m (x dx + y dy)

  • \sum m (p x + q y) dz = \sum (X y - Y x) - \sum (X_e y - Y_e x). $$

This equation, as well as the two other similar ones which would be obtained for virtual velocities of rotation about the other coordinate axes, does not simplify in general.

If the axis of rotation of the moving planes has a constant direction in space, in which case we know that it is the same with respect to the moving axes, we may take it for the axis of z, and we shall have

p=0,q=0. p = 0, \quad q = 0.

The above equation then becomes

∑m(yd2xdt2−xd2ydt2)−2r∑m(xdx+ydy)=∑(Xy−Yx)−∑(Xey−Yex). \sum m \left( y \frac{d^2x}{dt^2} - x \frac{d^2y}{dt^2} \right) - 2r \sum m (x dx + y dy) = \sum (X y - Y x) - \sum (X_e y - Y_e x).

Thus this relation is always equal in number to the equations of relative motion when the axis of rotation of the motion carrying the moving planes has a constant direction in space. If the moving points, in their relative motion, do not change distance with respect to the axis about which the areas are taken — that is to say, starting from which the coordinates x and y are here reckoned — the correction term 2r∑m(xdx+ydy)2r \sum m (x dx + y dy) disappears from the above equation.

If the forces XYXY are directed toward the origin of the coordinates, they disappear from this equation. It will be the same for the forces XeX_e , YeY_e , if the axis of rotation preserves a constant direction, whatever axis of zz is taken, and if the angular velocity of rotation of the moving planes is uniform — that is to say, if rr is constant and equal to ω\omega ; we shall then have

∑m(yd2xdt2−xd2ydt2)=2ω∑m(xdx+ydy). \sum m \left( y \frac{d^2x}{dt^2} - x \frac{d^2y}{dt^2} \right) = 2\omega \sum m (x dx + y dy).

By denoting by AA the variable moment of inertia of the system at any instant, and by λ\lambda the sum of the areas described on the plane of x,yx, y ; by integrating between two instants, and indicating the first limit by the index zero, we shall have

dλdt−dλ0dt=2ω(A−A0). \frac{d\lambda}{dt} - \frac{d\lambda_0}{dt} = 2\omega (A - A_0).

Thus, in the preceding hypotheses, the differentials of the areas in relative motions, when they are projected onto a plane perpendicular to the axis of rotation, increase like the moments of inertia.

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