Name: calculating cylinder moment of inertia
Uploaded: 2023-09-22T11:49:36.000Z
Duration: 1 min 13 s
Channel: JoVE

The moment of inertia of a wheel&#39;s axle is important to understand its behavior when rotating.
The axle can be approximated to a solid cylinder with constant density.
Consider an elemental disc parallel to the circular face of the cylinder at a certain distance from the perpendicular axis.
The mass moment of inertia for the disc about its diameter equals one-fourth the product of the mass and radius squared.
Applying the parallel-axis theorem, the moment of inertia of the disc with respect to the perpendicular axis can be estimated.
The differential mass equals the product of the cylinder&#39;s linear mass density and the disc&#39;s thickness.
Integrating the expression over the cylinder&#39;s length gives its moment of inertia about the perpendicular axis passing through the centroid.
For calculating the moment of inertia about the longitudinal axis, a thin cylindrical shell is considered.
Again, the differential mass is substituted into the moment of inertia equation and integrated over the cylinder&#39;s radius.
So, the moment of inertia about both these axes is a geometrical entity.

mass moment of inertia: problem solving

Mass Moment of Inertia: Problem Solving

Knowing how to determine the moment of inertia in a wheel's axle can be invaluable in engineering and automotive applications. It provides an understanding of how changes in geometry, mass, and radius can impact its performance.
The axle can be approximated to a solid cylinder with longitudinal and perpendicular axes. Initially, a thin disc is considered parallel to the circular face of the cylinder.
<img alt="Cylindrical symmetry physics diagram; axes labeled x, y, z; mass element dM, length L, radius R." src="/files/ftp_upload/14353/14353_Figure_1.svg" style="text-align: center; height: 225px;" data-altupdatedat="1761830439000" data-alt="Figure 1">
This disc has its moment of inertia equal to one-fourth the product of mass and radius squared. The disc is located at a certain distance from the perpendicular axis. This allows using the parallel-axis theorem, which estimates the moment of inertia of the disc about its perpendicular axis.
<img alt="Static equilibrium formula, dIy=1/4dMr²+dMx², diagram; analyzes rotational inertia concepts." src="/files/ftp_upload/14353/14353_PT_Equation_1v2.svg" style="text-align: center; height: 30px;" data-altupdatedat="1761830439000" data-alt="Equation 1">
The differential mass is expressed as the product of the cylinder's linear mass density and the disc's thickness.
<img alt="Differential calculus equation, dM=Md/L*dx, used in beam bending analysis, static equilibrium." src="/files/ftp_upload/14353/14353_PT_Equation_2v2.svg" style="text-align: center; height: 30px;" data-altupdatedat="1761830439000" data-alt="Equation 2">
The expression is then integrated over length to calculate the moment of inertia of the cylinder along the perpendicular axis through the centroid. 
<img alt="Moment of inertia equation diagram; static equilibrium formula, integral calculation." src="/files/ftp_upload/14353/14353_PT_Equation_3v2.svg" style="text-align: center; height: 33px;" data-altupdatedat="1761830439000" data-alt="Equation 3">
Furthermore, considering a thin cylindrical shell, the moment of inertia of the cylinder along its longitudinal axis can be estimated.
The differential mass of the shell element is substituted into the moment of inertia expression and integrated over the radius of the cylinder to get its moment of inertia along its longitudinal axis.
<img alt="Moment of inertia formula; integral and mathematical equations; physics concept diagram." src="/files/ftp_upload/14353/14353_PT_Equation_4v2.svg" style="text-align: center; height: 33px;" data-altupdatedat="1761830439000" data-alt="Equation 4">
The moment of inertia expressions for both cases depends on geometric factors like the cylinder's mass, radius and length. The moment of inertia for a cylinder is simply another geometrical parameter. This formula allows engineers and designers to better understand their measurements in designing new products or fixing older equipment.

Knowing how to determine the moment of inertia in a wheel&#39;s axle can be invaluable in engineering and automotive applications. It provides an ...

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Moment of Inertia

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calculating cylinder moment of inertia

Calculating the moment of inertia of a cylinder is important in engineering and automotive design. It helps show how changes in mass, radius, and shape can affect performance. For a wheel axle, a solid cylinder model can be used with both longitudinal and perpendicular axes.

One step in the calculation uses a thin disc drawn parallel to the circular face of the cylinder. The disc has a moment of inertia equal to one-fourth the product of mass and radius squared. The disc is then shifted away from the perpendicular axis, so the parallel-axis theorem can be used to find its moment of inertia about that axis.

The differential mass is written as the product of the cylinder's linear mass density and the disc's thickness. That expression is integrated over the length of the cylinder to get the moment of inertia about the perpendicular axis through the centroid. A second approach treats the cylinder as a thin cylindrical shell. The shell element is used in the moment of inertia expression, and the result is integrated over the radius to find the moment of inertia along the longitudinal axis.

Both moment of inertia formulas depend on geometry, including mass, radius, and length. In this way, the moment of inertia of a cylinder becomes another geometric parameter to measure and use. Engineers and designers can apply these results when creating new products or repairing older equipment.

Calculating Cylinder Moment of Inertia

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Calculating the moment of inertia of a cylinder is important in engineering and automotive design. It helps show how changes in mass, radius, and shape can affect performance. For a wheel axle, a solid cylinder model can be used with both longitudinal and perpendicular axes. One step in the calculation uses a thin disc drawn parallel to the circular face of the cylinder. The disc has a moment of inertia equal to one-fourth the product of mass and radius squared. The disc is then shifted away...

key terms and concepts

Learn how to calculate a cylinder’s moment of inertia using discs, shells, and the parallel-axis theorem for engineering and automotive applications.

Calculating Cylinder Moment of Inertia | JoVE K12

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Calculating Cylinder Moment of Inertia

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