CHAPTER VI.

MOTION IN A CIRCLE.

Explanation.—A particle in motion under the influence of a single force moves along a straight path. If two or more forces act upon it, their joint effect is to alter either the direction or velocity of the motion, or both. If the forces act along the same path as the particle, their joint effect is to accelerate or retard its motion, according as to whether their direction is the same as or opposite to that of the particle. If the forces act at an angle to the path of the particle, their joint effect is to change the original direction of the particle into another direction, which may be either a straight line or a curve. Under certain conditions the curve is a circle.

Fig. 115.

Uniform Motion in a Circle.-Let A, B, C, D (fig. 115) be points in the circular path described by the particle. Within the circle describe a polygon, whose sides, A B, B C, nearly approximate to the circle. Let A B be the path described by the particle in the first unit of time, A B will therefore represent the velocity during that time. Produce A B to E, and make BE= A B. If no other force acted upon the particle describing the path AB in the first unit of time, its path in the second unit of time would be BE. At B let another

force act upon the particle in the direction BD, at right angles to AB and equal to B F. This force will not alter the velocity, but only the direction of the motion of the particle, which at the end of the second second will arrive at C, instead of at E, and will describe the path B C. Complete the parallelogram BEC F, and join CD. BF will then represent the deflecting force, and B E the original force. The force BF is called the centripetal force, and is equal to the force CE, which tends to keep the particle away from the centre, and is therefore called the centrifugal force.

To determine the amount of the deflecting force, join A C. The two triangles DBC and B G C are similar, therefore DB: BC:: BCBG BC2=DB X B G. Let r=the radius, and ƒ the centripetal force. DB=2r, and BG=}BF={ƒ, and BC=v. By substituting these values we have

The centrifugal force is, therefore, equal and opposite to the deflecting or centripetal force, and consequently directly proportional to the square of the velocity, and inversely proportional to the radius of the circle described. In other words, the centrifugal force may be found by multiplying the weight of the body by the square of its velocity, and dividing by the acceleration of gravity and the radius of the circle.

An illustration of centrifugal force is shown in a stone whirled round by a string. If the stone be 1 pound weight, the string three feet long, and its velocity 24 feet per second, by the above formula, 1 lb. x 24 x 24÷32×3-6 lbs., the centrifugal force acting on the stone, which is counteracted by the string, representing the centripetal force, which is also equal to 6 pounds.

The centrifugal forces of two unequal bodies moving with equal velocities at different distances from the centre, are to one another as their quantities of matter multiplied by their distance from the

centre.

Effects of Centrifugal Force.-If a globe be turned about a fixed axis, each point in it describes the circumference of a circle, whose plane is perpendicular to the

1

L., centrum, the centre; peto, to seek. L., centrum; and fugo, to flv.

axis, and whose radius is equal to the distance of the point from the axis.

All points on the globe which describe unequal circles in equal times have velocities which are inversely proportional to the radii of the circles they describe.

The velocity of the particles composing the earth is greatest at the equator, and diminishes as we approach the poles, where the motion is nothing. The tendency of the particles to fly from the centre is, therefore, greatest at the equator; and if these particles were free to move, they would fly off. We have every reason to believe the earth was once in a fluid state,1 when the particles would possess a much smaller amount of cohesion than at present. The motion of the earth would cause the particles at the equator to move from the centre, while the particles at the poles would approach the centre.

This may be approximately illustrated by the following apparatus (fig. 116).

A, B, are two circular hoops of thin steel, fixed to the

centre at C, but capable of moving up and down the axis at D. E is a multiplying wheel, by which rapid motion may be communicated to the hoops.

As the velocity increases, the particles composing the hoops, being acted upon by centrifugal force, seek to fly off from the axis, but are prevented from so doing by the force of cohesion existing between them. The resultant motion of all the particles will cause the diameter of the hoops to extend in a direction perpendicular to the axis of rotation. The circumference of the hoops remaining unaltered in length, the diameter along the axis will decrease to allow for the increase of diameter occasioned by the centrifugal force.

The shape of the hoops when in motion is called an ellipse; and the figure generated by an ellipse moving about its shorter axis is called an oblate spheroid, which is the shape assumed by the earth.

The variation in the shape of the earth from a perfect sphere is one of the causes of variation in the force of gravity; and as the centrifugal force is greatest at the equator, and diminishes towards the poles, the tendency of a body to fly off is greatest at the equator; and as this tendency is opposed to the attraction of gravitation, its effect must, of course, be deducted from it. The force of gravity is, therefore, least at the equator, because it acts upon bodies which are farther away from the centre, and have greater centrifugal force.

It is easy to conceive that if the velocity were great enough, a revolving globe would assume a disc-like shape by the polar diameter becoming infinitely short. This effect is utilized in glass-blowing, where the blower takes up on the end of his blow-tube a ball of molten glass, and by whirling it round causes the particles to fly from the centre and thus form a circular plate of glass.

Centrifugal force is also illustrated by the trundling of a mop. The woollen fibres of the mop are fixed to the handle; but the water, being free to move, flies off.

Carriages on Curves.-The neglect of the effects of centrifugal force often causes carriages to overturn when rapidly driven round a curve. The weight of the carriage acts in the direction of the vertical, whether the carriage is in motion or at rest. When a motion round a centre is given to the carriage, a centrifugal force is generated, acting from the centre of gravity of the carriage at right angles to the axis of the curve.

Thus: If A (fig. 117) be the centre of gravity of the

B

Fig. 117.

Fig. 118.

carriage, its weight will act in the direction of the line AB. If the carriage be driven sharply round a curve, the centrifugal force A C will be generated, and the resultant of these two forces will be A D, falling outside the wheels, and the carriage will consequently be upset.

If the plane on which the body moves be inclined downwards towards the centre of the curve (fig. 118), the resultant will fall within the base, and the equilibrium will be stable. For this reason, in sharp railway curves, the outer rail is made higher than the inner. For the same reason a skater, when turning round, leans toward the centre of the curve he is describing.

The Governor.-One of the principal uses to which centrifugal force is applied, is to regulate the speed of machines, especially steam engines, where the resistance is variable. The mechanism exemplifying this, called a governor, is shown in fig. 119.

A B, A C, are two arms, capable of moving on axes at