Analytical mechanics for students of physics and engineering . ie in the x-axis. Thenthe mOSl Convenient element of mass is a strip which is parallel to the x-axis. Let ,/ be the length (Fig. 84), b thewidth, and <t the mass per unit area of the lamina, thendm = aady. I r :i physical definition of moment f inertia and its meaning see p. 220. Fig. 84. CENTER OF MASS AND MOMENT OF INERTIA 153 The distance of the element of mas-; from the axis is y\ therefore substi-tuting in equation (II) these expressions for dm and its distance from the axis we obtain I = ) //- • an dy = i (rub3 = i mb2, fo
Analytical mechanics for students of physics and engineering . ie in the x-axis. Thenthe mOSl Convenient element of mass is a strip which is parallel to the x-axis. Let ,/ be the length (Fig. 84), b thewidth, and <t the mass per unit area of the lamina, thendm = aady. I r :i physical definition of moment f inertia and its meaning see p. 220. Fig. 84. CENTER OF MASS AND MOMENT OF INERTIA 153 The distance of the element of mas-; from the axis is y\ therefore substi-tuting in equation (II) these expressions for dm and its distance from the axis we obtain I = ) //- • an dy = i (rub3 = i mb2, for the desired moment of inertia. The limits of integration arc differentfrom those in equation (II) because the independent variable is changedfrom m to y, 2. Find the moment of inertia about the x-axis of a lamina which isbounded by the parabola x2 = 2 py and the straight line y = a. (a) Choosing a horizontal strip for the element of mass we have dm = a - 2xdy. /axf-xdy0 = 2 a f y2V2py dyJ 0 >• But m = a I 2 x dy •J 0 = I a a v2 (b) We can also take an element of the strip for the element of mass,in which case we have dm = a dx dy. :. I = \ ir • a dx dy = i a a3 V2 pa,= 2 ma2. 154 AXAIATICAL MECHANICS PROBLEMS. 1. Find the moment of inertia of a circular lamina about a diameter. 2. Find the moment of inertia of an elliptical lamina about its minoraxis. 3. Bind the moment of inertia of a rectangular plate of negligiblethickness about a diagonal. 4. Find the moment of inertia of a thin plate, which is in the shape ofan equilateral triangle, with respect to one of its edges. 5. Find the moment of inertia of a triangular plate about an axiswhich i lasses through one of its vertices and is parallel to the base. 6. Find the moments of inertia of the following lamina with respectto the axes indicated by the thin vertical lines. a2b Ah b a2a b ; 1 1 J b 1 n 3ba 2a2b a 1 a 2b2a (a) (b) (c) (d) (e) 121. Theorems on Moments of Inertia. Theorem I. — Themoment of inertia
Size: 1656px × 1509px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, bookpublishernewyo, bookyear1913
