Derive the expression for the moment of inertia about the center of mass of a homogeneous...
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Derive the expression for the moment of inertia about the center of mass of a homogeneous rectangular plate of mass M(L+W²). 12 M, length L, and width W and show that I cm,plate In class we derived the moment of inertia about the center of mass of a homogeneous rod of mass Axis P 1 ML. As shown to the right, consider the plate as 12 M, length L and width W to be Im rod %3D cm M W being made up of an infinite number of differential rods, each of mass dM, length L, and width dy, and each having a moment of inertia about their own center of mass given by Iem = M (L?+ w?) 1 L'dM dl = 12 Moment of Inertia of a uniform ст plate rotating about an axis P through its cm. By using the Parallel Axis Theorem, each differential rod has a moment of +y inertia about the axis P given by dl, = dl m +h'dM where h is equal to y. With a homogeneous plate (i.e. mass density is constant), the mass dM of a rod is proportional to its area dA= Ldy as the total mass M is related to %3D y= W/2 dy dM +x the total Area (L·W). Thus, by summing up all dlp (i.e. integrating over), P the contributions of all rods Ip can be found. y = -W/2 Show all work. Derive the expression for the moment of inertia about the center of mass of a homogeneous rectangular plate of mass M(L+W²). 12 M, length L, and width W and show that I cm,plate In class we derived the moment of inertia about the center of mass of a homogeneous rod of mass Axis P 1 ML. As shown to the right, consider the plate as 12 M, length L and width W to be Im rod %3D cm M W being made up of an infinite number of differential rods, each of mass dM, length L, and width dy, and each having a moment of inertia about their own center of mass given by Iem = M (L?+ w?) 1 L'dM dl = 12 Moment of Inertia of a uniform ст plate rotating about an axis P through its cm. By using the Parallel Axis Theorem, each differential rod has a moment of +y inertia about the axis P given by dl, = dl m +h'dM where h is equal to y. With a homogeneous plate (i.e. mass density is constant), the mass dM of a rod is proportional to its area dA= Ldy as the total mass M is related to %3D y= W/2 dy dM +x the total Area (L·W). Thus, by summing up all dlp (i.e. integrating over), P the contributions of all rods Ip can be found. y = -W/2 Show all work.
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