Solid sphere Moment of inertial is a very important parameter in mechanics. If a rigid body rotates about the z-axis, the moment of inertial can be calculated (1) as a double integral (x² +y²)dA(x,y) if the rigid body is a 2D disk (left two figures), where Oo is the mass density per unit area; (2) as a triple integral II x +y)poav(x,y,z) if the rigid body is a 3D sphere (right figure), where pois the mass density per unit volume. Calculate the moment of inertial for the three rigid bodies: a 2D circular disk, a 2D circular ring, and a 3D sphere as shown in the figures. Express them in terms of total mass M and the geometric parameters (radii or radius). The total mass for 2D desks is M = oA, where A is the total area, and the total mass for 3D sphere is M=PoV, where V is the total volume.
Solid sphere Moment of inertial is a very important parameter in mechanics. If a rigid body rotates about the z-axis, the moment of inertial can be calculated (1) as a double integral (x² +y²)dA(x,y) if the rigid body is a 2D disk (left two figures), where Oo is the mass density per unit area; (2) as a triple integral II x +y)poav(x,y,z) if the rigid body is a 3D sphere (right figure), where pois the mass density per unit volume. Calculate the moment of inertial for the three rigid bodies: a 2D circular disk, a 2D circular ring, and a 3D sphere as shown in the figures. Express them in terms of total mass M and the geometric parameters (radii or radius). The total mass for 2D desks is M = oA, where A is the total area, and the total mass for 3D sphere is M=PoV, where V is the total volume.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Solid sphere
Moment of inertial is a very important parameter in mechanics. If a rigid body rotates about the z-axis, the moment of inertial can be calculated
(1). as a double integral
y)o,dA(x,y) if the rigid body is a 2D disk (left two figures), where Oo is the mass density per unit area;
(2) as a triple integral
I x2 +y)podV(x,y,z) if the rigid body is a 3D sphere (right figure), where po is the mass density per unit volume.
Calculate the moment of inertial for the three rigid bodies: a 2D circular disk, a 2D circular ring, and a 3D sphere as shown in the figures.
Express them in terms of total mass M and the geometric parameters (radii or radius). The total mass for 2D desks is M = 0oA, where A is the total area, and the
%3D
total mass for 3D sphere is M3POV. where V is the total volume.
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