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.

The moment of inertia of a solid sphere

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Secondly, we can sum up the moments of these exceedingly small thin disks along a given axis.

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If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Log in Sign up Search for courses, skills, and videos. Torque, moments, and angular momentum. Learn how the distribution of mass can affect the difficulty of causing angular acceleration. What is rotational inertia? Rotational inertia is a property of any object which can be rotated.

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. 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.

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Purchase Now. Want to know more about this Super Coaching? To calculate the moment of inertia of a solid sphere, let's consider a uniform solid sphere with mass M and radius R. Download as PDF. Report An Error. Important Links. How to calculate the moment of inertia of a solid sphere? Download as PDF. To do this, we need to slice the solid sphere into infinitely thin solid cylinders. 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. Important Links. Last updated on Jul 31,

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

What is the parallel axis theorem? How to calculate the moment of inertia of a solid sphere? Frequently Asked Questions What is the formula for the moment of inertia of a sphere? Test Series. What is the moment of inertia of a sphere about its axis on the surface? In the image shown above, r, x and R form a triangle. 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. Using the Pythagoras theorem, we can derive:. The higher the moment of inertia, the more torque is needed to change the angular velocity. 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. Inertia is a fundamental property of matter, which makes it resist any changes to its state of rest or motion. Frequently Asked Questions What is moment of inertia? 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 next step involves adding x into the equation.