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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The higher the moment of inertia, the more torque is needed to change the angular velocity. More Articles for Physics.

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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. It is a scalar value which tells us how difficult it is to change the rotational velocity of the object around a given rotational axis. Rotational inertia plays a similar role in rotational mechanics to mass in linear mechanics.

The moment of inertia of a solid sphere

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:.

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Important Links. The larger the moment of inertia the larger is the amount of torque required to cause any change in the angular velocity. Let's delve into the derivation process. The moment of inertia of a solid cylinder is given as:. Explore SuperCoaching Now. 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. The higher the moment of inertia, the more torque is needed to change the angular velocity. Purchase Now. Test Series. Report An Error. Download as PDF. What is the moment of inertia of a sphere about its axis on the surface? The moment of inertia of a sphere is a significant concept in physics. View Test Series.

In the preceding section, we defined the moment of inertia but did not show how to calculate it.

What is the parallel axis theorem? Want to know more about this Super Coaching? To calculate the moment of inertia of a solid sphere, the sphere is sliced into infinitesimally thin solid cylinders. Want to know more about this Super Coaching? In the image shown above, r, x and R form a triangle. Frequently Asked Questions What is the formula for the moment of inertia of a sphere? 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 the property of an object to resist any change in its state of rotatory motion. More Articles for Physics. The formula above specifically pertains to a solid sphere's moment of inertia about its central axis. It's often represented as follows:. More Articles for Physics. Using the Pythagoras theorem, we can derive:. Secondly, we can sum up the moments of these exceedingly small thin disks along a given axis.