SOLUTIONS OF THE EXAMPLES APPENDED TO A TREATISE ON THE MOTION OF A RIGID BODY. BY WILLIAM N. GRIFFIN, B.D., FELLOW AND TUTOR OF ST JOHN'S COLLEGE. CAMBRIDGE: J. DEIGHTON; M.DCCC.XLVIII. EXAMPLES OF THE MOTION OF A RIGID BODY. The references in the following solutions are to articles in the treatise which they are intended to follow. SECTION I. GEOMETRICAL PROPERTIES OF A RIGID BODY. 1. (a) IF x', y' be the co-ordinates of m referred to axes originating in the centre and parallel to those to which x, y refer, x = a + x', y = b + y'. .. Σ(mxy) = Σ(m) . ab + aΣ(my') + b Z (m x') + Z (m x'y'). Now Σ(mx') = 0} by properties of the centre of gravity. Also Σ(mx'y') = 0, since for any given value of y' the values of form pairs equal in magnitude and opposite in sign. Since .. (may) = Σ(m), ab = Mab. 0 for every point of the mass, The same method applies to (ẞ), (e), (Y) Since = 0 for every point of the mass, (7) Since & Σ(myx) = 0, Σ(mxx) = 0. Let OB = a, OA= b. (fig. 1). If P be a point of the lamina whose co-ordinates are x, y, the area of an element at P, whose sides are parallel to the axes of co-ordinates, is da.dy, and this may also represent the mass of a corresponding element of the lamina if the mass of an unit of area of the lamina be unity; .. Σ(mxy) = f f xy. Now first the summation of the values of this function for elements which lie along the line MQ and form an elementary strip of the body in that direction, is equivalent to integrating with respect to y from y = 0 to y = MQ = (a − x) tan B. Hence for such a strip Σ(myx) becomes = 1⁄2 ƒ (a3a - Secondly, the required summation will be completed by adding the values of the function for such strips as those just considered, as they range from OA to B, and this amounts to integrating the expression just obtained from a = 0 to x = a; .. finally, (may) = ↓ (} − } + 4) a' tan B = 24 a2b2; while M, the mass, has by virtue of the units adopted been represented by ab; (d) Let l, m, n be the cosines of the inclinations of this line to the co-ordinate axes; r the distance of a point x, y, z in the line from its centre, and let the mass of an element at this point be represented by its length dr. The co-ordinates of the centre of the line being = 1 M {2 (be + b'c') + be' + b'e}. So Σ(mxx) = } M {2 (ac + a'c') + ac′ + a'c}. Σ(mxy) = 1M {2 (ab + a'b') + ab + a'b}. (n) This is an instance where the evaluation of a function is expedited by transforming it to other co-ordinate axes. Let x', y be the co-ordinates of the point x, y, when referred to other rectangular axes in the plane of xy so that a' is in the axis of the cone; The facility of the limits of integration in a' and y′ recommends them in preference to x and y. Changing the integration to the former, we have Σ(mxy) = √ √ xy = dx dy dx dy -S, Jay (dx dy dy dz (Gregory's Ex. Chap. 111., or Moigno, Vol. 11. p. 214) |