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. 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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Explore SuperCoaching Now. The moment of inertia of a solid cylinder is given as:.

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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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To do this, we need to slice the solid sphere into infinitely thin solid cylinders. To calculate the moment of inertia of a solid sphere, let's consider a uniform solid sphere with mass M and radius R. Report An Error. Calculating the Moment of Inertia of a Solid Sphere To calculate the moment of inertia of a solid sphere, let's consider a uniform solid sphere with mass M and radius R. More Articles for Physics. Understanding the Moment of Inertia of a Rectangle - Testbook. The next step involves adding x into the equation. The moment of inertia is calculated around an axis passing through its center. Explore SuperCoaching Now. 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 sphere is a significant concept in physics.

The moment of inertia of a solid cylinder is given as:. It's often represented as follows:. Important Links. Report An Error. 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. Test Series. Secondly, we can sum up the moments of these exceedingly small thin disks along a given axis. In this equation, R signifies the radius of the sphere and M represents its mass. Important Links. Want to know more about this Super Coaching?