Question

1. A thin, circular children's toy has mass and dimensions as n the adjacent figure. Neglect the mass of the struts (parts with hatched symbol). (a) It is rotating about its central axis which is perpendicalar to the plane of the object. Calculate the moment of inertia for rotation about this axis (b) It is then made to spin on the end of a child's finger so that it splns, about an axis which is perpendicular to the plane of the object but is half way between ri and ry. Calculate the moment of inertia for rotation about this axis (c) The child then spins the object on the end of thelr finger so hat rotates vertically, that Is about an axis, whlch is in te plane of the object, passlng through the centre of mass. Calculate the moment of inertia for rotation about this axis

1. A thin, circular children's toy has mass and dimensions as n the adjacent figure. Neglect the mass of the struts (parts with hatched symbol). (a) It is rotating about its central axis which is perpendicalar to the plane of the object. Calculate the moment of inertia for rotation about this axis (b) It is then made to spin on the end of a child's finger so that it splns, about an axis which is perpendicular to the plane of the object but is half way between ri and ry. Calculate the moment of inertia for rotation about this axis (c) The child then spins the object on the end of thelr finger so hat rotates vertically, that Is about an axis, whlch is in te plane of the object, passlng through the centre of mass. Calculate the moment of inertia for rotation about this axis

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Answer 1

r_1 = inner radius r_2 = outer radius rho = mass per unit area of ring. we consider a ring of radius x from centre with width dx. moment of inertia of the ring w.r.to the axis = mass of the ring times x^2 = 2 pi x dx rho x^2 = 2 pi x^3 rho dx Area = pi (x + dx)^2 - pi x^2 = 2 pi x dx [dx^2 is neglected] Now: integrating betn r_1 to r_2: Therefore I = 2 pi rho integral_r_1^r_2 x^3 dx = 2 pi rho/4 (r_2^4 - r_1^4) Now: M = mass of the object = pi rho (r_2^2 - r_1^2) = pi rho/2 (r_2^2 + r_1^2) (r_2^2 - r_1^2) I = M/2 (r_2^2 + r_1^2) Axis is in the plane of the object at passing through the center of mass: (Diameter core) Let, M.I about diameter = I_2. Here: we use perpendicular axis theorem:

a and c is done here in the 1st page with diagram besides

this is b part