# What is Moment of Inertia?

Moment of inertia, also known as rotational inertia, is an important concept in physics that describes the inertia of an object when it rotates. Just as mass in linear motion affects the motion of an object, moment of inertia affects the rotation of an object.

Definition of moment of inertia

The moment of inertia is the inertia of an object as it rotates and depends on the mass and shape of the object, as well as the distribution of the mass relative to the axis of rotation. Formulaically, the moment of inertia I can be expressed as: I = ∑mr², where m is the mass of the object and r is the distance from the mass to the axis of rotation.

The moment of inertia reveals the inertial characteristics of an object during rotation, that is, the resistance of the object. When an object is acted upon by an external torque, its moment of inertia determines the rate of change of its rotational state. According to the definition of moment of inertia, we can draw the following conclusions:

The greater the mass of an object, the greater its moment of inertia.

The closer the object is distributed to the axis of rotation, the smaller its moment of inertia is.

Effect of moment of inertia

The moment of inertia determines the dynamic properties of an object as it rotates. An object with a larger moment of inertia requires more torque to change its rotational speed, while an object with a smaller moment of inertia requires less torque. This is why ice hockey players tighten their arms as they spin: it reduces their moment of inertia, thus increasing their spin speed.

Calculation of moment of inertia

For some common configurations of objects and rotational axes, we can use standard formulas to calculate the moment of inertia. For example, for a uniform disk with mass m and radius r, if the axis of rotation passes through its center and is perpendicular to the disk surface, its moment of inertia is I=1/2mr².

Application of moment of inertia in engineering

Moment of inertia has many applications in engineering. For example:

Balance in mechanical design: Rotational inertia affects vibration and stability in mechanical systems. In order to balance a mechanical system, it is sometimes necessary to consider the moment of inertia in the design. For example, in rotating machinery, the moment of inertia is balanced by adjusting the mass distribution of the rotating parts. Moment of inertia is also important in motor design because it affects the motor's start and stop times.

Spacecraft attitude control: The spacecraft needs to control its attitude in space to maintain the correct trajectory and direction. Rotational inertia plays a key role here, and engineers need to consider the rotational inertia characteristics of the spacecraft to design a suitable attitude control scheme.

Automotive engineering: The design of automobile suspension systems needs to consider the moment of inertia to achieve good ride comfort and handling stability. For example, the rotational inertia can be balanced by adjusting the component distribution and stiffness in the suspension system, thereby improving the driving stability of the car.

Robotics: Robots need to move flexibly, and the moment of inertia will affect the motion control of the robot. When engineers design robots, they need to consider the impact of rotational inertia on the robot's motion performance and optimize the design to improve the robot's motion efficiency and stability.

Inertia of thin rod

When the axis of rotation passes through the midpoint (center of mass) of the rod and is perpendicular to the rod, the inertia is:

When the axis of rotation passes through the end point of the rod and is perpendicular to the rod, the inertia is:

m: the mass of the rod

L: the length of the rod

Inertia of cylinder

When the axis of rotation is the axis of the cylinder, the inertia is：

m: the mass of the cylinder

r: the radius of the cylinder

Inertia of thin ring

When the axis of rotation passes through the center of the ring and is perpendicular to the ring surface, the inertia is:

When the axis of rotation passes through the edge of the ring and is perpendicular to the ring surface, the inertia is:

When the axis of rotation is along a certain diameter of the ring, the inertia is:

m: the mass of the thin ring

R: the radius of the thin ring

Inertia of thin disk

When the rotation axis passes through the center and is perpendicular to the disk surface, the inertia is:

When the axis of rotation passes through the edge and is perpendicular to the disk surface, the inertia is:

m: the mass of the thin disk

R: the radius of the thin disk

Inertia of hollow cylinder

When the rotation axis is the symmetry axis of the hollow cylinder, the inertia is:

m: the mass of the hollow cylinder

R1 and R2: the inner and outer radii of the hollow cylinder respectively.

Inertia of spherical shell

When the rotation axis is the central axis of the spherical shell, the inertia is:

When the axis of rotation is the tangent line of the spherical shell, the inertia is:

m: the mass of the spherical shell

R: the radius of the spherical shell

Inertia of solid sphere

When the axis of rotation is the central axis of the sphere, the inertia is:

When the axis of rotation is the tangent to the sphere, the inertia is:

m: the mass of the sphere

R: the radius of the sphere

Inertia of the cube

When the rotation axis is the central axis of the cube, the inertia is:

When the axis of rotation is the edge of the cube, the inertia is:

When the axis of rotation is the diagonal of the cube, the inertia is:

m: the mass of the cube

L: the side length of the cube

Inertia of the cuboid

When the rotation axis is the central axis of the cuboid, the inertia is:

m: the mass of the cuboid

l1 and l2: the lengths of the two sides of the rectangle perpendicular to the axis of rotation

Definition of moment of inertia

The moment of inertia is the inertia of an object as it rotates and depends on the mass and shape of the object, as well as the distribution of the mass relative to the axis of rotation. Formulaically, the moment of inertia I can be expressed as: I = ∑mr², where m is the mass of the object and r is the distance from the mass to the axis of rotation.

The moment of inertia reveals the inertial characteristics of an object during rotation, that is, the resistance of the object. When an object is acted upon by an external torque, its moment of inertia determines the rate of change of its rotational state. According to the definition of moment of inertia, we can draw the following conclusions:

The greater the mass of an object, the greater its moment of inertia.

The closer the object is distributed to the axis of rotation, the smaller its moment of inertia is.

Effect of moment of inertia

The moment of inertia determines the dynamic properties of an object as it rotates. An object with a larger moment of inertia requires more torque to change its rotational speed, while an object with a smaller moment of inertia requires less torque. This is why ice hockey players tighten their arms as they spin: it reduces their moment of inertia, thus increasing their spin speed.

Calculation of moment of inertia

For some common configurations of objects and rotational axes, we can use standard formulas to calculate the moment of inertia. For example, for a uniform disk with mass m and radius r, if the axis of rotation passes through its center and is perpendicular to the disk surface, its moment of inertia is I=1/2mr².

Application of moment of inertia in engineering

Moment of inertia has many applications in engineering. For example:

Balance in mechanical design: Rotational inertia affects vibration and stability in mechanical systems. In order to balance a mechanical system, it is sometimes necessary to consider the moment of inertia in the design. For example, in rotating machinery, the moment of inertia is balanced by adjusting the mass distribution of the rotating parts. Moment of inertia is also important in motor design because it affects the motor's start and stop times.

Spacecraft attitude control: The spacecraft needs to control its attitude in space to maintain the correct trajectory and direction. Rotational inertia plays a key role here, and engineers need to consider the rotational inertia characteristics of the spacecraft to design a suitable attitude control scheme.

Automotive engineering: The design of automobile suspension systems needs to consider the moment of inertia to achieve good ride comfort and handling stability. For example, the rotational inertia can be balanced by adjusting the component distribution and stiffness in the suspension system, thereby improving the driving stability of the car.

Robotics: Robots need to move flexibly, and the moment of inertia will affect the motion control of the robot. When engineers design robots, they need to consider the impact of rotational inertia on the robot's motion performance and optimize the design to improve the robot's motion efficiency and stability.

Inertia of thin rod

When the axis of rotation passes through the midpoint (center of mass) of the rod and is perpendicular to the rod, the inertia is:

When the axis of rotation passes through the end point of the rod and is perpendicular to the rod, the inertia is:

m: the mass of the rod

L: the length of the rod

Inertia of cylinder

When the axis of rotation is the axis of the cylinder, the inertia is：

m: the mass of the cylinder

r: the radius of the cylinder

Inertia of thin ring

When the axis of rotation passes through the center of the ring and is perpendicular to the ring surface, the inertia is:

When the axis of rotation passes through the edge of the ring and is perpendicular to the ring surface, the inertia is:

When the axis of rotation is along a certain diameter of the ring, the inertia is:

m: the mass of the thin ring

R: the radius of the thin ring

Inertia of thin disk

When the rotation axis passes through the center and is perpendicular to the disk surface, the inertia is:

When the axis of rotation passes through the edge and is perpendicular to the disk surface, the inertia is:

m: the mass of the thin disk

R: the radius of the thin disk

Inertia of hollow cylinder

When the rotation axis is the symmetry axis of the hollow cylinder, the inertia is:

m: the mass of the hollow cylinder

R1 and R2: the inner and outer radii of the hollow cylinder respectively.

Inertia of spherical shell

When the rotation axis is the central axis of the spherical shell, the inertia is:

When the axis of rotation is the tangent line of the spherical shell, the inertia is:

m: the mass of the spherical shell

R: the radius of the spherical shell

Inertia of solid sphere

When the axis of rotation is the central axis of the sphere, the inertia is:

When the axis of rotation is the tangent to the sphere, the inertia is:

m: the mass of the sphere

R: the radius of the sphere

Inertia of the cube

When the rotation axis is the central axis of the cube, the inertia is:

When the axis of rotation is the edge of the cube, the inertia is:

When the axis of rotation is the diagonal of the cube, the inertia is:

m: the mass of the cube

L: the side length of the cube

Inertia of the cuboid

When the rotation axis is the central axis of the cuboid, the inertia is:

m: the mass of the cuboid

l1 and l2: the lengths of the two sides of the rectangle perpendicular to the axis of rotation

Updated on: 31/10/2023

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