Physics

## HUYGENS

# Theorems on Centrifugal Force

**I.**

If two equal moving bodies pass over unequal circumferences in equal times, the centrifugal force in the greater circumference will be to that in the smaller, as the circumferences are to each other, or as their diameters.

**II.**

If two equal moving bodies move with equal velocities in unequal circumferences, their centrifugal forces will be in the inverse ratio of the diameters.

**III.**

If two equal moving bodies move in equal circumferences with unequal velocities, but each of them with uniform motion, which we are to understand is the case in all these propositions, the centrifugal force of the more rapidly moving body will be to that of the more slowly moving body in the squared ratio of the velocities.

**IV.**

If two equal moving bodies moving in unequal circumferences have equal centrifugal force, the time of describing the greater circumference will be to the time of describing the smaller circumference in the ratio of the square roots of the diameters.

**V.**

If a moving body travels in the circumference of a circle with a velocity the same as that which it would acquire by falling from a height equal to a quarter of the diameter, it will have a centrifugal force equal to its weight; that is, it will stretch the string which holds it to the center with the same force as it would if suspended from it.

**VI.**

If a moving body travels in circumferences parallel to the horizon, drawn on the inner surface of a parabolic conoid, the axis of which is perpendicular, all the circumferences, whether they are small or great, will be described in equal times; and each of these times is equal to the time of a double oscillation of a pendulum, the length of which is half that of the latus rectum of the generating parabola.

**VII.**

If two moving bodies, suspended by threads of unequal length, swing around so that they traverse circumferences parallel to the horizon, while the other end of the thread remains fixed, and so that the altitudes of the cones which the threads describe in this motion are equal, then the times in which their paths are described are equal.

**VIII.**

If two moving bodies, as before, swing around so as to describe cones, and are suspended by threads either equal or unequal, and if the altitudes of the cones are unequal, the times of revolution will be in the ratio of the square roots of their altitudes.

**IX.**

If a pendulum which swings around in a cone describes exceedingly small circuits, the times of describing each of them will have the same ratio to the time of perpendicular fall through a distance equal to twice the length of the pendulum as the circumference of a circle has to its diameter; and further, will be equal to the time of two lateral oscillations of the same pendulum, if they are very small.

**X.**

If a body moves around in a circumference and completes each circuit in the same time as that in which a pendulum, the length of which is equal to the radius of this circumference, will complete a very small circuit when describing a cone, or two very Small lateral oscillations, then it will have a centrifugal force equal to its weight.

**XI.**

The time of revolution of any pendulum which describes a cone will be equal to the time of perpendicular fall through a height equal to the length of the pendulum, if the angle of inclination of the thread to the horizontal plane is approximately 2 degrees 54 minutes. The exact statement is: If the sine of the angle described is to the radius as the square inscribed in a circle is to the square on its circumference.

**XII.**

If two pendulums equal in weight but with threads of unequal length swing around as conical pendulums, and the altitudes of the cones are equal, then the forces by which the threads are stretched are in the same ratio as the lengths of the threads.

**XIII.**

If a simple pendulum swings with its greatest lateral oscillation, that is, if it descends through the whole quadrant of a circle, when it comes to the lowest point of the circumference it stretches the string with three times as great a force as it would if it were simply suspended by it.