744 MECHANICS Flexure the case of a uniform plank clamped horizontally at one end, ^l. ! ailk and otherwise unsupported. This is obviously the same as the case of a plank of double the length, supported by a trestle placed under its middle. We assume as before that the radius of curvature is always very large compared with the thickness of the plank. In all such cases we may at once apply the principle of 258, and suppose one portion of the plank up to a section F to be fixed in its equilibrium position. The curvature immediately contiguous to P will then be simply propor tional to the moment about P of the forces acting on the unfixed portion. Hence at the free end there will be no curvature, and the curvature at points near that end will be of the second order of infinitesimals; i.e., its rate of increase at the end vanishes. Let x be the length of the fixed portion, I the whole length of the plank. Then, as the deflexion y from the horizontal is always very small, the curvature is expressed ( 22) by d-y dx* so that we have at once where E is the "flexural rigidity " of the plank, and ^uits mass per unit of length. Successive integrations give d E?/= A - B(Z - x) - 7jV<7(^ - a-) 4 , The terminal conditions are for #=0, 2/ = 0, -r- = 0: dx aiidfora-7, ^f = 0, ^ = 0. The last two are obviously satisfied. The two former give B = ,-, A= El + ?z/j.gl* = - zftgl* . Hence Ey -stuff (31* - 4Z s (Z-a;) + (l-sf) 4 ) . Thus the droop of the free extremity (x = l) is E~ = ~8E where AV is the whole weight. If the plank had been weightless, but loaded at the free end with
- v weight AV, our equation would have been
and we should have had The terminal conditions at rc = are as before, so that B = - J AV/- 1 , A = - -i VP + 1 W< = - i W7, and the dmop of the free end is ^ , greater than before in the ratio of 8 : 3. If the plank be again looked on as heavy, but its free end be sup ported on a trestle which is pressed upwards till it acts with a force W, we find directly E// = A" - 15" (/ - ,r) The terminal conditions, at x = 0, are still as in the first case, and they give B"-tWP-J^P, -.vhen the amount by which the free end is raised is This is obviously the same as the amount of depression of the middle of a plank of length 21 supported by trestles at each end. 276. Hence the droop of the middle of a plank resting on trestles at its ends is to that of the ends when the plank re-sts on a single trestle at the middle in the ratio of 5 : 3. If the equation expressing the curvature in the first or third cases above be twice differentiated, the common result is // v>*i J The simplicity of this expression leads us to seek for the most general form. Suppose the plank to be exposed to any system of forces in lines perpendicular to its length and breadth. Then, if any transverse section be made, the stress between the two portions of the plank will con sist of forces ( + G) and couples ( + H) in the plane of length and thickness. Let the applied forces be N per unit of length. Suppose also, as before, that the radius of curvature is very great compared with the thickness. Then the equations of equilibrium of an element are -j- t +N-0, f j- + G = Q. We have also the condition of bending, viz., rc-E??-] dx- Eliminating H and G among these equations, we have E x curvature = E , , = H . dx- fa 4 dx 1 dx " which of course includes all the previous particular cases. We may now determine (under the limits imposed) the form of a uniform plank of any length, supported in a nearly horizontal position at different points in its length, and loaded at any assigned points with any weights. The importance of this in practice is obvious. 277. But we may easily take a further step, and in- v vestigate the oscillatory motion, so long at least as the* 1 acceleration parallel to the length of the plank and its rota- 1 j tion are negligible. For in such a case, if //. be the mass * per unit of length, the equation of motion is ( 199) We will consider only the case in which the applied force N may be neglected. This is practically the case of a uniform wire or flat rectangular spring. (Suppose, further, that it is fixed at one end and free at the other, like Wheatstone s "kaleidophone," or like the tongue of a reed organ-pipe. Then, writing n 4 for the fraction /x/E, we have A particular integral may obviously be found in the form y = r]cos(i~t/n-+ a) ...... (1), where 77 (a function of a-) and i/n (a constant number) have to be found ; o is any constant. The substitution of this value of y leads to the complete integral of which is i] = Ae ! * + Bg - + C cos (ix + D) . Now, provided the value of i be properly determined, the motion represented by (1), with the above value of ij can exist by itself; and the most general motion of which the spring is capable (under the limits imposed) consists of super position of a number of separate motions of a similar character. Hence this may be treated by itself. Our limiting conditions in the present case are and n " 0> e&" at the fixed end
= ~ = at tlie free cn< *
