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. Similarly, the moment of inertia is an object's ability to resist changes to its rotational motion.

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

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In the image shown above, r, x and R form a triangle. Important Links.

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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.

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. 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. We start by considering the moment of inertia of a thin disc, which is given as:. The next step involves adding x into the equation. Considering the diagram, we can see that r, R, and x form a triangle. Using the Pythagorean theorem, we get:.

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Secondly, we can sum up the moments of these exceedingly small thin disks along a given axis. The moment of inertia is the property of an object to resist any change in its state of rotatory motion. The formula above specifically pertains to a solid sphere's moment of inertia about its central axis. Frequently Asked Questions What is moment of inertia? Download as PDF. Want to know more about this Super Coaching? Inertia is a fundamental property of matter, which makes it resist any changes to its state of rest or motion. It's often represented as follows:. Report An Error. The next step involves adding x into the equation. Purchase Now. Similarly, the moment of inertia is an object's ability to resist changes to its rotational motion.

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. This section is very useful for seeing how to apply a general equation to complex objects a skill that is critical for more advanced physics and engineering courses.

Important Links. View Test Series. The moment of inertia is the property of an object to resist any change in its state of rotatory motion. Using the Pythagorean theorem, we get:. To calculate the moment of inertia of a solid sphere, the sphere is sliced into infinitesimally thin solid cylinders. Frequently Asked Questions What is moment of inertia? Download as PDF. Similarly, the moment of inertia is an object's ability to resist changes to its rotational motion. Torque is the force that makes an object rotate around an axis. Explore SuperCoaching Now. Inertia is a fundamental property of matter, which makes it resist any changes to its state of rest or motion. Test Series. View Test Series. Want to know more about this Super Coaching? Understanding the Parallel Axis Theorem The parallel axis theorem is a fundamental principle used in physics to calculate the moment of inertia of a rigid body about any axis, given the moment of inertia about the parallel axis through the body's center of mass and the perpendicular distance between the axes.