Determine the mass moment of inertia of the assembly about the z axis. The density of the materi

Determine the mass moment of inertia of the assembly about...

Key Concepts

Axis of Rotation

The axis of rotation, such as the z-axis in this context, is critical in defining how the mass distribution is measured when computing the moment of inertia. Different axes yield different moment of inertia values for the same body, making it important to properly understand and specify the axis about which calculations are made.

Mass Moment of Inertia

The mass moment of inertia is a measure of an object's resistance to angular acceleration about a given axis. It quantifies how the mass of a body is distributed relative to that axis, and it is crucial in determining the rotational dynamics of the system. This concept is fundamental in both theoretical and applied mechanics, as it directly influences the response of a body to applied torques.

Mass Density

Mass density refers to the mass per unit volume of a material, and it plays a pivotal role in calculating the mass and mass distribution within a body. In the context of moment of inertia, density is used to relate the geometric distribution of a body with its mass, thereby allowing the integration of mass elements over the volume to determine the overall inertia.

Volume Integration for Inertia Calculations

Volume integration is used to compute the mass moment of inertia for continuous bodies by summing up the contributions of infinitesimal mass elements. Each element contributes to the inertia according to its mass and the square of its distance from the axis of rotation. This method is essential for accurately capturing the role of geometry and density in the calculation of the inertia for complex shapes.

Transcript

00:01 Hi everyone here we have to find the moment of inertia of the assembly about z axis.

00:08 The density of material is given 7 .85 megagram per meter cube or it can be written as 7 .85 10 to the power 3 kg per meter cube.

00:26 Let us see here the solution given assembly is the composition of three segments.

00:57 Segment 1 cone of radius 0 .3 meter and height of 0 .67 meter let us see here this is the segment 1 segment 2 is hemisphere of radius 0 .3 meter and third segment is negative mass of the system which has to be removed from the given system it is a cone of radius 0 .1 meter and height of point 2 to 5 having negative mass of the system because it has to be removed now mass of each segment having the value density into volume density is given 7 .85 10 to the power 3 kg per meter cube first tagments is a cone so its volume will be one third of pi r square h substituting the value radius is given 0 .3 height is given 0 .67 so m1 b will get 158 point 9625 5 pi kg mass m2 hemispair so it having the value 2 by 3 pi r2 cube into row volume into density radius is 0 .3 density 7 .85 10 to the power 3 so hemisphere having the mass 141 .3 kg mass of m3 having the negative value volume into density density is 7 .85 10 to the power 3 volume is 1 third of pi r square h because it is a cone so from here we can write a radius is 0 .1 height is 0 .2 .5 so you will get m3 to be negative of 5 .88 7 5 by kg now moment of inertia of the given assembly will be submission of moment of inertia of each segment about the z axis so i z1 will be 3 by 10 m1 r1 square i j2 is called to 2 by 5 m2 r2 square and i z 3 3 minus 3 by 10 m 3 r3 square substituting the value you will get ij to be 3 by 10 m1 r1 square 2 by 5 m 2 r2 square 3 by 10 m 3 r2 square substituting the value mass m1 is given 158 096255 5 5 5 5 5 5 5 and radius is 0 .3.

06:49 Mass of hemispair 141...

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