The moment of inertia of a solid sphere

Inertia is a fundamental property of matter, which makes it resist any changes to its state of rest or motion. The more significant the inertia, the greater the force needed to alter its velocity within a specific time frame.

The moment of inertia of a sphere is a significant concept in physics. It's often represented as follows:. In this equation, R signifies the radius of the sphere and M represents its mass. The formula above specifically pertains to a solid sphere's moment of inertia about its central axis. However, if we consider the sphere's moment of inertia about an axis on its surface, the expression changes to:. Let's delve into the derivation process.

The moment of inertia of a solid sphere

.

The larger the moment of inertia the larger is the amount of torque required to cause any change in the angular velocity. Report An Error.

.

Analytical formulas for the moments of inertia second moments of area I x , I y and the products of inertia I xy , for several common shapes are referenced in this page. The considered axes of rotation are the Cartesian x,y with origin at shape centroid and in many cases at other characteristic points of the shape as well. Also, included are the formulas for the Parallel Axes Theorem also known as Steiner Theorem , the rotation of axes, and the principal axes. The moment of inertia, or more accurately, the second moment of area, is defined as the integral over the area of a 2D shape, of the squared distance from an axis:. From the definition, it is apparent that the moment of inertia should always have a positive value, since there is only a squared term inside the integral. Conceptually, the second moment of area is related with the distribution of the area of the shape. Specifically, a higher moment, indicates that the shape area is distributed far from the axis. On the contrary, a lower moment indicates a more compact shape with its area distributed closer to the axis. For example, in the following figure, both shapes have equal areas, whereas, the right one, features higher second moment of area around the red colored axis, since, compared to the left one, its area is distributed quite further away from the axis. More than often, the term moment of inertia is used, for the second moment of area, particularly in engineering discipline.

The moment of inertia of a solid sphere

The moment of inertia of a sphere is a significant concept in physics. It's often represented as follows:. In this equation, R signifies the radius of the sphere and M represents its mass. The formula above specifically pertains to a solid sphere's moment of inertia about its central axis.

Monkeys clipart

The next step involves adding x into the equation. Purchase Now. The higher the moment of inertia, the more torque is needed to change the angular velocity. View Test Series. Important Links. What is the moment of inertia of a sphere about its axis on the surface? Report An Error. Explore SuperCoaching Now. Frequently Asked Questions What is moment of inertia? Understanding the Moment of Inertia of a Rectangle - Testbook.

In the preceding subsection, we defined the moment of inertia but did not show how to calculate it. In this subsection, we show how to calculate the moment of inertia for several standard types of objects, as well as how to use known moments of inertia to find the moment of inertia for a shifted axis or for a compound object.

The higher the moment of inertia, the more torque is needed to change the angular velocity. The moment of inertia of a sphere is derived by slicing the solid sphere into infinitesimally thin solid cylinders and then summing the moments of these exceedingly small thin disks. Test Series. Torque is the force that makes an object rotate around an axis. How is the moment of inertia of a sphere derived? Understanding the Moment of Inertia of a Rectangle - Testbook. The moment of inertia of a solid cylinder is given as:. However, if we consider the sphere's moment of inertia about an axis on its surface, the expression changes to:. Let's delve into the derivation process. It's often represented as follows:. Important Links. The moment of inertia is the property of an object to resist any change in its state of rotatory motion.