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:.

The moment of inertia of a solid sphere

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The moment of inertia of a sphere is a significant concept in physics.

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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 moment of inertia of a solid sphere

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.

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In this equation, R signifies the radius of the sphere and M represents its mass. It's often represented as follows:. More Articles for Physics. View Test Series. Want to know more about this Super Coaching? Purchase Now. The larger the moment of inertia the larger is the amount of torque required to cause any change in 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. 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. Explore SuperCoaching Now. How to calculate the moment of inertia of a solid sphere?

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What is the moment of inertia of a sphere about its axis on the surface? Using the Pythagorean theorem, we get:. Test Series. The moment of inertia of a solid cylinder is given as:. 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. More Articles for Physics. Explore SuperCoaching Now. Last updated on Jul 31, 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. The moment of inertia is calculated around an axis passing through its center. The moment of inertia of a sphere is a significant concept in physics. To calculate the moment of inertia of a solid sphere, let's consider a uniform solid sphere with mass M and radius R. Frequently Asked Questions What is moment of inertia? View Test Series.