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Let

F=0

.(1)

be the equation to the given surface, z being the vertical axis.

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to

From the condition that t, is to be a minimum we obtain

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But Sx and Sy are not independent, (1) gives us

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which, by means of (1), may be reduced to a differential equation of the second order between two variables; the integral will therefore contain two arbitrary constants, which will enable us to make the curve pass through the two given points.

202. A particle acted on by any forces, and resting on a smooth horizontal plane, is attached by an inextensible string

to a point which moves in a given manner in that plane; to determine the motion of the particle.

Let x, y, x, y be the co-ordinates, at time t, of the particle and point, a the length of the string, and R the force of constraint.

For the motion of the particle we have

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with the condition (x-x)+(y-3)=a2. Now, are given functions of t.

Take from both
d'x d'y
de re-

sides of the equations in (1) the quantities

dt2

dt2

spectively, and we have the equations of relative motion

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These are precisely the equations we should have had if the point had been fixed, and in addition to the forces X, Y and R acting on the particle, we had applied, reversed in direction, the accelerations of the point's motion. It is evident that the same theorem will hold in three dimensions. The dx dy accelerations are known as functions of t, and theredt, dt dt

fore the equations of relative motion are completely determined. Compare § 24.

203. Let there be no impressed forces, and suppose first that the point moves uniformly in a straight line.

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introduced in the equations of motion. We have thus the case of § 26.

Again, suppose the point's motion to be rectilinear, but uniformly accelerated.

The relative motion will evidently be that of a simple pendulum from side to side of the point's line of motion. In certain cases, when the angular velocity exceeds a certain limit, we shall have the string occasionally untended; and this will give rise to an impact (Chap. X.) when it is again tended. While the string is untended the particle moves, of course, in a straight line.

204. Suppose the point to move, with uniform angular velocity w, in a circle whose radius is r and center origin. Here, supposing the point to start from the axis of x,

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or, in polar co-ordinates, for the relative motion,

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Now

wt is the inclination of the string to the radius.

passing through the point; call it ø, and we have

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which is the ordinary equation of motion of a simple pendulum

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The particle therefore moves, with reference to the uniformly revolving radius of the circle described by the point, just as a simple pendulum with reference to the vertical.

205. To determine the motion of a particle acted on by given forces, and constrained to move in a smooth tube, in the form of a given plane curve, of indefinitely small sectional area, which revolves in a given manner about an axis in its plane.

Let the axis of revolution be that of z, and let the position. of the particle at time t be given by its distance r from that axis, the plane of the tube at that instant making an angle with a fixed plane passing through the axis. By the conditions of the problem is a given function of t.

The sole effect of the tube will be to produce a force of constraint, which lies in the normal plane to the tube, and may therefore be resolved into two parts, one perpendicular to the plane of the tube, the other in that plane and in the principal normal to the tube.

Let the impressed forces be resolved into three, Palong r, T perpendicular to the plane of the tube, and S parallel to the axis of z.

Let R, R' be the two resolved parts of the force of constraint.

The equations of motion will then be (by §§ 15, 64)

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where s is the arc of the revolving curve.

In addition to these we have the two equations

0 = f (t)

which gives the position of the tube at any time, and

the equation to the tube.

r = $(z)

(4),

(5),

By means of (4) and (5) we may eliminate 0, r, and s from (1), (2), (3). Then eliminating R between (1) and (3), we obtain a differential equation between and t, whose integral together with (4) completely determines the position of the particle at any instant.

R and R' may then be found from (1) or (3), and (2).

In general the angular velocity of the tube is given conde

stant, or =w, whence() becomes 0=wt if the plane from

dt

which is measured be that of the tube at the time t=0.

The simplest case we can take is the following.

206. A particle moves in a smooth straight tube which revolves uniformly round a vertical axis to which it is perpendicular, to determine the motion.

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