Moment of inertia of a disk formula

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# Moment of inertia of a disk formula

In the preceding section, we defined the moment of inertia but did not show how to calculate it. In this section, 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. Because r is the distance to the axis of rotation from each piece of mass that makes up the object, the moment of inertia for any object depends on the chosen axis. 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 in the center of the barbell, each of the two masses m is a distance R away from the axis, giving a moment of inertia of. 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. Figure 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. However, this is not possible unless we take an infinitesimally small piece of mass dmas shown in Figure. 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:.

This, in fact, is the form we need to generalize the equation for complex shapes. It is best to work out specific examples in detail to get a feel for how to calculate the moment of inertia for specific shapes. This is the focus of most of the rest of this section. Consider a uniform density and shape thin rod of mass M and length L as shown in Figure.

Physics - Application of the Moment of Inertia (3 of 11) Solid Cylinder Rolling Down an Incline

We want a thin rod so that we can assume the cross-sectional area of the rod is small and the rod can be thought of as a string of masses along a one-dimensional straight line. In this example, the axis of rotation is perpendicular to the rod and passes through the midpoint for simplicity. Our task is to calculate the moment of inertia about this axis. We orient the axes so that the z -axis is the axis of rotation and the x -axis passes through the length of the rod, as shown in the figure.

This is a convenient choice because we can then integrate along the x -axis. We define dm to be a small element of mass making up the rod. The moment of inertia integral is an integral over the mass distribution. However, we know how to integrate over space, not over mass.The moment of inertia of a thin circular disk is the same as that for a solid cylinder of any length, but it deserves special consideration because it is often used as an element for building up the moment of inertia expression for other geometries, such as the sphere or the cylinder about an end diameter.

The moment of inertia about a diameter is the classic example of the perpendicular axis theorem For a planar object:. Moment of Inertia: Thin Disk The moment of inertia of a thin circular disk is the same as that for a solid cylinder of any length, but it deserves special consideration because it is often used as an element for building up the moment of inertia expression for other geometries, such as the sphere or the cylinder about an end diameter.

Index Moment of inertia concepts. Superposition of Moments of Inertia The moment of inertia of a composite object can be obtained by superpositon of the moments of its constituent parts.

The Parallel axis theorem is an important part of this process. The composite moment of inertia is given by the sum of the contributions shown at left. Principal Axes The use of Newton's second law for rotation involves the assumption that the axis about which the rotation is taking place is a principal axis. Since most common rotational problems involve the rotation of an object about a symmetry axis, the use of this equation is usually straightforward, because axes of symmetry are examples of principle axes.

A principal axis may be simply defined as one about which no net torque is needed to maintain rotation at a constant angular velocity. The issue is raised here because there are some commonly occurring physical situations where the axis of rotation is not a principal axis. For example, if your automobile has a tire which is out of balance, the axle about which it is rotating is not a principal axis. Consequently, the tire will tend to wobble, and a periodic torque must be exerted by the axle of the car to keep it rolling straight.

At certain speeds, this periodic torque may excite a resonant wobbling frequency, and the tire may begin to wobble much more violently, vibrating the entire automobile.Angular momentum relates to how much an object is rotating. An object has a constant angular momentum when it is neither speeding up nor slowing down.

The angular momentum of an object depends on the distribution of the mass of the object. The moment of inertia is a value that describes the distribution. It can be found by integrating over the mass of all parts of the object and their distances to the center of rotation, but it is also possible to look up the moments of inertia for common shapes.

The angular momentum is the product of the moment of inertia and the angular velocity around an axis. The moment of inertia of a solid disc iswhere M is the mass of the disc, and R is the radius.

When a DVD in a certain machine starts playing, it has an angular velocity of What is the angular momentum of this disc?

Answer: The angular momentum can be found using the formula, and the moment of inertia of a solid disc ignoring the hole in the middle. The angular momentum is:. The moment of inertia of a hollow sphere iswhere M is the mass and R is the radius. If the basketball weighs 0. Answer: The angular momentum of the basketball can be found using the moment of inertia of a hollow sphere, and the formula. Toggle navigation.Moment of inertia I is defined as The sum of the products of the mass of each particle of the body and square of its perpendicular distance from the axis.

It is also known as rotational inertia. The moment of inertia reflects the mass distribution of a body or a system of rotating particles, with respect to an axis of rotation. The moment of inertia only depends on the geometry of the body and the position of the axis of rotation, but it does not depend on the forces involved in the movement.

For a rigid body moving about a fixed axis, the laws of motion have the same form as those of rectilinear motion, with the moment of inertia replacing mass, angular replacing linear velocity, angular momentum replacing linear momentum, etc.

Consider a mass m attached to the end of a massless rod. Let us assume that the bearing at the pivot point O is frictionless. Let the system be in a horizontal plane. A force F is acting on the mass perpendicular to the rod and hence this will accelerate the mass according to:. In doing so the force will cause the mass to rotate about O. Since tangential acceleration is related to angular.

### 10.6: Calculating Moments of Inertia

This can be done by multiplying both sides of the above equation by r. The moment of inertia plays the same role in angular motion as the mass in linear motion. Your email address will not be published. Save my name, email, and website in this browser for the next time I comment. Facebook Twitter Pinterest. Related Articles. Examples of Mechanical Energy list 4 weeks ago. Difference Between Speed and Velocity with Examples 3 weeks ago. Difference between Speed And Acceleration 4 weeks ago.

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Close Search for.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:. This, in fact, is the form we need to generalize the equation for complex shapes. It is best to work out specific examples in detail to get a feel for how to calculate the moment of inertia for specific shapes. This is the focus of most of the rest of this section.

We want a thin rod so that we can assume the cross-sectional area of the rod is small and the rod can be thought of as a string of masses along a one-dimensional straight line. In this example, the axis of rotation is perpendicular to the rod and passes through the midpoint for simplicity. Our task is to calculate the moment of inertia about this axis. We orient the axes so that the z-axis is the axis of rotation and the x-axis passes through the length of the rod, as shown in the figure.

## Moment of Inertia Formulas

This is a convenient choice because we can then integrate along the x-axis. We define dm to be a small element of mass making up the rod. The moment of inertia integral is an integral over the mass distribution. However, we know how to integrate over space, not over mass. We therefore need to find a way to relate mass to spatial variables. Since the mass density of this object is uniform, we can write. We chose to orient the rod along the x-axis for convenience—this is where that choice becomes very helpful.

The distance of each piece of mass dm from the axis is given by the variable x, as shown in the figure.For example, if two disks have the same mass but one has all the mass around the rim and the other is solid, then the disks would have different moments of inertia. Calculating moments of inertia is fairly simple if you only have to examine the orbital motion of small point-like objects, where all the mass is concentrated at one particular point at a given radius r.

Here, r is the radius of the circle, from the center of rotation to the point at which all the mass of the golf ball is concentrated. Crunching the numbers can get a little sticky when you enter the non—golf ball world, however, because you may not be sure of which radius to use.

### Rotational-Linear Parallels

When you have an extended object, such as a rod, each bit of mass is at a different radius. You can use this concept of adding up the moments of inertia of all the elements to get the total in order to work out the moment of inertia of any distribution of mass. Say you have two golf balls, and you want to know what their combined moment of inertia is.

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If you have a golf ball at radius r 1 and another at r 2the total moment of inertia is. So how do you find the moment of inertia of, say, a disk rotating around an axis stuck through its center? You have to break the disk up into tiny balls and add them all up. You complete this using the calculus process of integration. The figure depicts the shapes that these moments of inertia correspond to.

The shapes corresponding to the moments of inertia in the table.Moment of Inertia has the same relationship to angular acceleration as mass has to linear acceleration.

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For a point mass the Moment of Inertia is the mass times the square of perpendicular distance to the rotation reference axis and can be expressed as. The Moment of Inertia with respect to rotation around the z-axis of a single mass of 1 kg distributed as a thin ring as indicated in the figure above, can be calculated as. Point mass is the basis for all other moments of inertia since any object can be "built up" from a collection of point masses. For rigid bodies with continuous distribution of adjacent particles the formula is better expressed as an integral.

The Radius of Gyration is the distance from the rotation axis where a concentrated point mass equals the Moment of Inertia of the actual body. The Radius of Gyration for a body can be expressed as. Moments of Inertia for a thin-walled hollow cylinder is comparable with the point mass 1 and can be expressed as:. Moments of Inertia for a slender rod with axis through center can be expressed as. Moments of Inertia for a slender rod with axis through end can be expressed as.

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