Thus, in the Ex. § 37, the angular acceleration is

RRR

41. The motion of a point in a plane being given with respect to fixed axes, to investigate expressions for its velocity and acceleration relative to axes in the same plane, which revolve about a common origin with constant angular velocity. The revolving Let @ be this angular velocity; then, if at time t = 0 the fixed and revolving axes coincide, at time t they will be res are inclined to one another at an angle wt. Hence, if x, y, §, n be When lobe the co-ordinates of the point at time t, referred to the fixed and to the revolving axes respectively, we have by the ordinary formulæ for transformation of co-ordinates

ectangular.

which determine the velocities relative to the revolving axes.

d2

Again,

=

dÆ
dt2 dt2

d3n d2y

dt

or

-=

dt

dt

'cos wt) - w

dt

(3),

dy

dt

dt

Now the component accelerations along fixed axes, with which at the time t the moving axes coincide, are evidently represented by the first two terms of the right-hand sides of these equations; or, in terms of the co-ordinates with respect to the moving axes, by

Ex. If the point be at rest, x and y are constant, and

These expressions are obvious, as in this case the relative motion of the point with respect to the moving axes is a uniform circular motion about the origin, in the negative direction, i.e. from the axis of n to that of §.

R 42. Suppose the new axes not to revolve uniformly.

In this case the investigation is precisely the same as the above, with the exception that e, a given function of t, must be substituted for wt. If w, now no longer constant, be put do for the student will have no difficulty in verifying the

dt

following expressions, which take the place of (2), (3′) and (4) of the preceding section.

These expressions might have been deduced at once from the expressions in § 16, by the consideration of relative accelerations as in § 26. Let OM, MP=n, be the co-ordinates of the point referred to the moving axes. Then, by § 16, the acceleration of M along OM is

d°E
dt2

Also, as MP revolves with angular velocity w, the acceleration of P relative to M, in the direction perpendicular to MP, is

This is in the direction of the negative part of the axis of . Hence the resolved part parallel to Og, of the acceleration of P with respect to O, is

43. The principles already enunciated, and the examples given of their application, will suffice for the solution of problems on this part of the subject.

Other examples of the application of these principles, such as the kinematical part of the investigations of the Hodograph, &c., will be more appropriately introduced in future chapters.

EXAMPLES.

(1) A point moves from rest in a given path, and its velocity at any instant is proportional to the time elapsed since its motion commenced; find the space described in a given time.

(2) If a point begin to move with velocity v, and at equal intervals of time a velocity u be communicated to it in the same direction; find the space described in n such intervals.

(3) A man six feet high walks in a straight line at the rate of four miles an hour away from a street lamp, the height of which is 10 feet; supposing the man to start from the lamp-post, find the rate at which the end of his shadow travels, and also the rate at which the end of his shadow separates from himself.

(4) If the position of a point moving in a plane be determined by the co-ordinates p and 4, p being measured from a fixed circle (radius a) along a tangent which has revolved through an angle & from a fixed tangent; investigate the following expressions for the accelerations along and perpendicular to p respectively,

(5) Prove that it is not possible for a point to move so that its velocity in any position may be proportional to the length of the path which it has described from rest: also that if its velocity be proportional to the space it has to describe, however small, it will never accomplish it.

(6) The velocity of a point parallel to each of three rectangular axes is proportional to the product of the other. two co-ordinates; what are the equations of the path, and what is the time of describing a given portion when the curve passes through the origin?

(7) A point moves in a plane, so that its velocities • parallel to the axes of x and y are

u + ey and v + ex respectively, shew that it moves in a conic section.

(8) Two points are moving with constant velocity in two straight lines, 1st in a plane, 2nd in space; given the initial circumstances, find when they are nearest to each other. Shew also that in both cases the relative path is a straight line, described with constant velocity.

(9) A number of points are moving with constant velocity in straight lines in space; determine the motion of their common centre of inertia. (§ 58.)

(10) A cannon-ball is moving in a direction making an acute angle with a line drawn from the ball to an observer; if V be the velocity of sound, and nV that of the ball, prove that the whizzing of the ball at different points of its course will be heard in the order in which it is produced, or in the reverse order, according as n <> sec 0.

(11) A particle, projected with a velocity u, is acted on by a force, which produces a constant acceleration ƒ, in the plane of motion, inclined at a constant angle a to the direction of motion. Obtain the intrinsic equation of the curve described, and shew that the particle will be moving in the opposite direction to that of projection at the time

(12) Shew that any infinitely small motion given to a plane figure in its own plane is equivalent to a rotation through an infinitely small angle about some point in the figure.

Hence shew that the relative motion of two figures in a plane may be produced by rolling a curve fixed to one figure on a curve fixed to the other figure. (These curves called Centroids.)

are

(13) The highest point of the wheel of a carriage rolling on a road moves twice as fast as each of two points in the rim whose distance from the ground is half the radius of the wheel.

(14) A rod of given length moves with its ends in two given lines which intersect; shew how to draw a tangent to the path described by any point of the rod.

(15) Investigate the position of the instantaneous centre about which the rod is turning, and apply this also to solve the preceding question.

(16) One circle rolls on another whose centre is fixed. From the initial and final positions of a diameter in each