. Graphical and mechanical computation . sin a dx. Moment of area 1 C „ , 1 Cn . , , P C, s , „.,, = - / y2 dx = - I P sin2 a dx = - I (1 — cos 2 a) dx XX 2JJ 2 J 4 J I2 C I2 c = - I dx /sin (900 — 2 a) dx = /sin (900 — 2 a) dx, since / dx = o, about the arm PQ returning to its original position when P makes a completecircuit of the curve. Moment of inertiaof area about X ,,_ = - / -y3 dx = - I P sin3 adx = - I (- sin a sin 3 a )dx pf- p r = - I sin a dx / sin 3 a dx. 4J 12J Art. 108 INTEGRATORS 251 Now, / sin a dx, j sin (900 — 2 a) dx, and I sin 3 a dx, and hence the area, moment, and moment
. Graphical and mechanical computation . sin a dx. Moment of area 1 C „ , 1 Cn . , , P C, s , „.,, = - / y2 dx = - I P sin2 a dx = - I (1 — cos 2 a) dx XX 2JJ 2 J 4 J I2 C I2 c = - I dx /sin (900 — 2 a) dx = /sin (900 — 2 a) dx, since / dx = o, about the arm PQ returning to its original position when P makes a completecircuit of the curve. Moment of inertiaof area about X ,,_ = - / -y3 dx = - I P sin3 adx = - I (- sin a sin 3 a )dx pf- p r = - I sin a dx / sin 3 a dx. 4J 12J Art. 108 INTEGRATORS 251 Now, / sin a dx, j sin (900 — 2 a) dx, and I sin 3 a dx, and hence the area, moment, and moment of inertia can be measured by three in-tegrating wheels whose axes at any instant make angles a, 900 — 2 a,and 3 a, respectively, with XX. The Amsler yivheel integrator (Fig. 108c) consists of an arm PQ and3 integrating wheels A, M, and /. The instrument is guided by a carriagewhich rolls in a straight groove in a steel bar; this bar may be set at aproper distance from the hinge of the tracer arm by the aid of trams. The. Fig. 108c. line XX, which passes through the points of the trams and under thehinge, is the axis about which the moment and moment of inertia aremeasured. The radius of the disk containing the M-wheel is one-halfthe radius, and the radius of the disk containing the /-wheel is one-thirdthe radius of the circular disk D to which they are geared. Therefore,the axis of the M-wheel turns through twice, and the axis of the /-wheelturns through three times the angle through which the tracer arm PQ orthe axis of the A -wheel swings from the axis XX. The integrating wheels are set so that in the initial position, , whenP lies on XX, the axes of the A- and /-wheels are parallel to XX whilethe axis of the M-wheel is perpendicular to XX. Then, when the tracerarm PQ makes an angle a with XX, the axes of the A-, M-, and /-wheelsmake angles a, 900 — 2 a, and 3 a, respectively, with XX. Further-more, the graduations of the M-wheel are marked so that these
Size: 1915px × 1304px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubj, booksubjectengineering
