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Ex. 5. An imperfectly elastic ball is projected from a point in the circumference of a circle, and after twice rebounding from the circle returns to the same point again; shew that the direction of projection makes an angle a with the radius drawn to the point of projection which is given by the equation

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CHAPTER III.

ON UNIFORM ACCELERATING FORCES AND GRAVITY.

ACCORDING to the definition of a uniform accelerating force in Chapter I., the velocity generated by it in the same time is always the same, and by the second law of motion, is unaffected by the previous motion of the body. If we put ƒ=the force, measured by the velocity it generates in a unit of time, we shall have the velocity generated in t units=ft. Writing for this v velocity acquired by the body at the end of the time t from rest, we have therefore

v=ft

9. PROP. To find the relations of the space, time, and force when a body moves from rest under the action of a uniform accelerating force.

The velocity of the body is continually increased from 0 up to ft, if t be the time and ƒ the force. Let s be the whole space described in the time t, and let t be divided into n equal intervals, each

t

n

The velocities at the end of the times

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Now, if the body moved uniformly during each interval of time with the velocity it had at the beginning of the interval, from the expression space=velocity × time, we should have the whole space s equal to the sum of this series:

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If the body had moved uniformly during each interval of time with the velocity it had at the end of the interval, we should have s equal the sum of this series :

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n2

n to

n2

=ƒ—2 (1 + 2 + 3 + &c.

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Since the velocity is continually accelerated, the true value of s will be between these two quantities, however small each interval may be, or however great n may be; but when n is indefinitely great, the last terms in each of the above expressions vanish, and we have therefore

s=1 ft2

10. Between the two equations v=ft, and s=ft, we may eliminate either for t, and thus obtain

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The expression s=vt shews us that the space described from rest by the action of a uniform accelerating force is one half of the space which would have been described in the same time if the velocity had been constant and equal to its value at the end of the time.

If we put t=1 in the equation s=1ƒť, we have f=2s; or,

f, which is the velocity generated by the force in a unit of time, is measured by twice the space through which the body falls in a unit of time. It is found that a heavy body in our latitudes falls through a space of nearly 161 feet in the first second of time; therefore, if we put g = the accelerating force of gravity, we have g=velocity of 32.2 feet per second of time, or, with the understanding that one second is our unit of time, we write g=32.2 feet.

This value of the force of gravity is only an approximate value for small heights above the earth's surface.

Sir Isaac Newton's law of universal gravitation is, that every particle of matter attracts every other particle with a force which varies directly as the mass of the attracting particle, and inversely as the square of the distance. It is also shewn that a spherical body equally dense at equal distances from its center attracts a particle outside its surface as if the matter of the sphere were collected at its center; so that, considering the earth such a sphere, if g be the force of gravitation at the surface, ƒ the force at any point exterior to the surface at a distance r from the center, and R be earth's radius, we have

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μ in this expression is called the absolute accelerating force, for it is the value of ƒ when r=1. When R is taken for unity, r must be expressed in terms of the earth's radius, and μ=g= 32.2 feet.

When r is very near R, or for small heights above the earth's surface, we have f=g very nearly, or we may take gravity as a constant accelerating force in that case. It is found that, on account of the diurnal rotation of the earth on its axis, its figure differs sensibly from spherical, being flattened at the poles and bulging at the equator, or is an oblate spheroid. The centri

fugal force (see Article 31), being produced by the diurnal rotation, is greatest at the equator, and is there directly opposed to the force of gravity. It is also nothing at the poles. The resultant gravitation of a heavy body is affected by the direct action of the centrifugal force and its indirect action through the change of the figure of the earth. The ratio for the equator and poles is as follows:

gravitation at the equator: gravitation at the pole :: 186: 187. We can, for these reasons, only consider gravity as constant for the same latitude on the earth's surface, and for small altitudes above it. The direction of gravity at each point on the earth's surface being perpendicular to the surface taken as that of still water, does not pass accurately through the center of the oblate spheroid.

11. PROP. A body being projected with a given velocity u in the direction in which a uniform accelerating force f acts; to find its velocity, and the space passed over in a given time.

If v be the velocity, s the space described at the end of the time t, we shall have, by the second law of motion,

v=velocity of projection + velocity from the action of the force =u±ft

where the upper sign is to be taken when the force accelerates the velocity, and the lower when it retards it.

In the same way,

=

space described space due to velocity of projection+the space due to the action of the force

or, s=utft

the upper and lower signs to be taken as before.

12. PROP. A body being projected with a given velocity u in the direction in which a uniformly accelerating force facts; to find its velocity when it has passed through a given space.

Let v be the velocity when the body has passed through the space s.

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