Name: calculating rotation resistance
Uploaded: 2023-04-30T11:01:10.000Z
Duration: 1 min 9 s
Channel: JoVE

The moment of inertia of an object consisting of discrete mass elements is the sum of the product of the elements&#39; masses and the square of their distances from the axis of rotation.
For example, if two spheres of mass m and 2m are attached to the ends of a rod of negligible mass, at a distance R from its center, the moment of inertia about an axis passing through its center is equal to thrice mR squared.
If the particles making up an object are not discrete but continuous infinitesimal masses, the summation is replaced by an integral over the object.
For example, consider the mass M of a rod of length L uniformly distributed along a straight line. To calculate its moment of inertia, divide M into infinitesimally small pieces of mass dm and integrate the resulting quantity.
The moment of inertia about an axis fixed at one of its ends is then obtained to be equal to one-third of ML squared.

moment of inertia: calculations

Moment of Inertia: Calculations

The moment of inertia of an object depends on its axis of rotation. Consider a barbell consisting of two masses attached to the ends of a rod. The masses at the end can be considered point masses, and the rod can be assumed to have negligible weight. The moment of inertia of the barbell can be calculated using the mathematical definition of the moment of inertia once the system&#39;s rotation axis is decided. Suppose the point masses m&#160;are fixed at a distance R from the center of the rod, the moment of inertia about an axis passing through its center is equal to 2mR2 and about an axis at one of its ends, it is 4mR2. This suggests that it is twice as hard to rotate the barbell about the end than about its center.
The definition of the moment of inertia of a rigid body is derived from considering each element of mass at a fixed distance from the axis of rotation. However, for a rigid body built of continuous mass, each element is infinitesimal. Hence, the summation is replaced by the integral over infinitesimal mass elements.
In special cases, when the mass is distributed evenly throughout the object of interest, the mass element can be written as density multiplied by the element of length, area, or volume and other relevant constants. For example, imagine a thin uniform rod with mass uniformly distributed throughout its length. By relating the infinitesimal mass element with an infinitesimal length element via the constant linear mass density, its moment of inertia can be calculated. It is found that a uniform rod&#39;s moment of inertia about an axis fixed at its end is four times more than its moment of inertia about an axis passing through its center.
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/10-5-calculating-moments-of-inertia">Openstax, University Physics Volume 1, Section 10.5: Calculating Moments of Inertia.</a>

The moment of inertia of an object depends on its axis of rotation. Consider a barbell consisting of two masses attached to the ends of a rod. The masses ...

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Rotation and Rigid Bodies

KEY TERMS AND CONCEPTS

calculating rotation resistance

Moment of inertia measures how hard it is to rotate an object, and the answer depends on the axis of rotation. A simple barbell shows this clearly. It has two masses at the ends of a rod. The masses can be treated as point masses, and the rod can be assumed to have negligible weight.

Once the rotation axis is chosen, the moment of inertia can be found from the mathematical definition. If the two point masses each have mass m and are a distance R from the center of the rod, the moment of inertia about an axis through the center is 2 mR^2. About an axis through one end, it is 4 mR^2. That means the barbell is twice as hard to rotate about the end as it is about the center.

For a rigid body with continuous mass, the calculation uses an integral instead of a sum. Each piece of mass is infinitesimal, so the mass elements are added over the whole object. When the mass is spread evenly, the mass element can be written using density and an element of length, area, or volume, along with the needed constants.

A thin uniform rod is a useful example of this method. Its mass is evenly distributed along its length, so its infinitesimal mass can be related to an infinitesimal length using linear mass density. Using that setup, the rod’s moment of inertia about an axis at one end is four times its moment of inertia about an axis through its center.

Calculating Rotation Resistance

angular velocity and displacement

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Rotational Inertia and Energy Storage

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Moment of inertia measures how hard it is to rotate an object, and the answer depends on the axis of rotation. A simple barbell shows this clearly. It has two masses at the ends of a rod. The masses can be treated as point masses, and the rod can be assumed to have negligible weight. Once the rotation axis is chosen, the moment of inertia can be found from the mathematical definition. If the two point masses each have mass m and are a distance R from the center of the rod, the moment of...

moment of inertia of compound objects

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parallel-axis theorem

Moment of Inertia with Parallel Axes

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Moment of Inertia for a Flat Disc

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key terms and concepts

See how moment of inertia changes with the axis of rotation using a barbell and a uniform rod. Learn the key calculation idea for point and continuous masses.

Calculating Rotation Resistance | JoVE K12

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Calculating Rotation Resistance

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