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Calculate the moment of inertia of a thin plate, in theshape of a right triangle, about an axis that passes throughone end of the hypotenuse and is parallel to the oppositeleg of the triangle, as in Figure P10.28a. Let M representthe mass of the triangle and L the length of the base of thetriangle perpendicular to the axis of rotation. Let h representthe height of the triangle and w the thickness of theplate, much smaller than L or h. Do the calculation ineither or both of the following ways, as your instructorassigns:(a) Use Equation 10.17. Let an element of mass consistof a vertical ribbon within the triangle, of width dx,height y, and thickness w. With x representing the locationof the ribbon, show that y ” hx/L. Show that thedensity of the material is given by . ” 2M/Lwh. Showthat the mass of the ribbon is dm ” .yw dx ” 2Mx dx/L2.Proceed to use Equation 10.17 to calculate the momentof inertia.(b) Let I represent the unknown moment of inertiaabout an axis through the corner of the triangle. Notethat Example 9.15 demonstrates that the center of massof the triangle is two thirds of the way along the length L,from the corner toward the side of height h. Let ICM representthe moment of inertia of the triangle about an axisthrough the center of mass and parallel to side h.Demonstrate that I ” ICM ) 4ML2/9. Figure P10.28bshows the same object in a different orientation.Demonstrate that the moment of inertia of the triangularplate, about the y axis is Ih ” ICM ) ML2/9. Demonstratethat the sum of the moments of inertia of the trianglesshown in parts (a) and (b) of the figure must be the momentof inertia of a rectangular sheet of mass 2M andlength L, rotating like a door about an axis along its edgeof height h. Use information in Table 10.2 to write downthe moment of inertia of the rectangle, and set it equal tothe sum of the moments of inertia of the two triangles.Solve the equation to find the moment of inertia of a triangleabout an axis through its center of mass, in termsof M and L. Proceed to find the original unknown I.

Calculate the moment of inertia of a thin plate, in theshape of a right triangle, about an axis that passes throughone end of the hypotenuse and is parallel to the oppositeleg of the triangle, as in Figure P10.28a. Let M representthe mass of the triangle and L the length of the base of thetriangle perpendicular to the axis of rotation. Let h representthe height of the triangle and w the thickness of theplate, much smaller than L or h. Do the calculation ineither or both of the following ways, as your instructorassigns:(a) Use Equation 10.17. Let an element of mass consistof a vertical ribbon within the triangle, of width dx,height y, and thickness w. With x representing the locationof the ribbon, show that y ” hx/L. Show that thedensity of the material is given by . ” 2M/Lwh. Showthat the mass of the ribbon is dm ” .yw dx ” 2Mx dx/L2.Proceed to use Equation 10.17 to calculate the momentof inertia.(b) Let I represent the unknown moment of inertiaabout an axis through the corner of the triangle. Notethat Example 9.15 demonstrates that the center of massof the triangle is two thirds of the way along the length L,from the corner toward the side of height h. Let ICM representthe moment of inertia of the triangle about an axisthrough the center of mass and parallel to side h.Demonstrate that I ” ICM ) 4ML2/9. Figure P10.28bshows the same object in a different orientation.Demonstrate that the moment of inertia of the triangularplate, about the y axis is Ih ” ICM ) ML2/9. Demonstratethat the sum of the moments of inertia of the trianglesshown in parts (a) and (b) of the figure must be the momentof inertia of a rectangular sheet of mass 2M andlength L, rotating like a door about an axis along its edgeof height h. Use information in Table 10.2 to write downthe moment of inertia of the rectangle, and set it equal tothe sum of the moments of inertia of the two triangles.Solve the equation to find the moment of inertia of a triangleabout an axis through its center of mass, in termsof M and L. Proceed to find the original unknown I.

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