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 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 moment of inertia of a solid sphere

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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. In this case, the summation over the masses is simple because the two masses at the end of the barbell can be approximated as point masses, and the sum therefore has only two terms. In the case with the axis at the end of the barbell—passing through one of the masses—the moment of inertia is. From this result, we can conclude that it is twice as hard to rotate the barbell about the end than about its center. In this example, we had two point masses and the sum was simple to calculate. However, to deal with objects that are not point-like, we need to think carefully about each of the terms in the equation. Recall that in our derivation of this equation, each piece of mass had the same magnitude of velocity, which means the whole piece had to have a single distance r to the axis of rotation. The need to use an infinitesimally small piece of mass dm suggests that we can write the moment of inertia by evaluating an integral over infinitesimal masses rather than doing a discrete sum over finite masses:.

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.

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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. Report An Error. What is the moment of inertia of a sphere about its axis on the surface? Torque is the force that makes an object rotate around an axis. Inertia is a fundamental property of matter, which makes it resist any changes to its state of rest or motion. The moment of inertia is the property of an object to resist any change in its state of rotatory motion. The more significant the inertia, the greater the force needed to alter its velocity within a specific time frame. The higher the moment of inertia, the more torque is needed to change the angular velocity. 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. How is the moment of inertia of a sphere derived? To do this, we need to slice the solid sphere into infinitely thin solid cylinders. Frequently Asked Questions What is moment of inertia? The parallel axis theorem is a fundamental theorem in classical mechanics that is used to calculate the moment of inertia of a rigid body about any axis, given the body's moment of inertia about the parallel axis through the object's center of mass and the perpendicular distance between the axes. Similarly, the moment of inertia is an object's ability to resist changes to its rotational motion. Purchase Now.

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.

Important Links. Important Links. Test Series. Deriving the Moment of Inertia of a Sphere We can derive the moment of inertia of a sphere in two primary ways: Firstly, we can slice the solid sphere into infinitesimally thin solid cylinders. In this equation, R signifies the radius of the sphere and M represents its mass. The higher the moment of inertia, the more torque is needed to change the angular velocity. However, if we consider the sphere's moment of inertia about an axis on its surface, the expression changes to:. Test Series. Using the Pythagoras theorem, we can derive:. Explore SuperCoaching Now. Let's delve into the derivation process.