# Moment of inertia of a rod along its length

The concept might not seem too interesting at first, but in combination with the law of the conservation of angular momentum, it can be used to describe many fascinating physical phenomena and predict motion in a wide range of situations.

The moment of inertia for an object describes its resistance to angular acceleration, accounting for the distribution of mass around its axis of rotation. Rotational inertia works just like this for rotational motion, and the mathematical definition always includes mass.

### 10.6: Calculating Moments of Inertia

The same object can have multiple moments of inertia, however, because while a big part of the definition is about the distribution of mass, it also accounts for location of the axis of rotation.

The symbol I is used for moment of inertia, and it has units of kg m 2. In all moments of inertia, the term MR 2 appears, although for different shapes there are different fractions in front of this term, and in some cases there may be multiple terms summed together.

The MR 2 component is the moment of inertia for a point mass at a distance R from the axis of rotation, and the equation for a specific rigid body is built up as a sum of point masses, or by integrating an infinite number of small point masses over the object. While in some cases it may be useful to derive the moment of inertia of an object based on a simple arithmetic sum of point masses or by integrating, in practice there are many results for common shapes and axes of rotation that you can simply use without needing to derive it first:.

Solid cylinder central diameter axis, or the diameter of the circular cross-section in the middle of the cylinder :.

Understanding why there are different equations for each axis of rotation is a key step to grasping the concept of a moment of inertia. Think about a pencil: You can rotate it by spinning it around in the middle, by the end or by twisting it around its central axis. Because the rotational inertia of an object depends on the distribution of mass about the axis of rotation, each of these situations is different and requires a separate equation to describe it.

You can get an instinctive understanding of the concept of moment of inertia if you scale this same argument up to a foot flag pole. Spinning it end over end would be very difficult — if you could manage it at all — whereas twirling the pole about its central axis would be much easier. This is because torque depends strongly on the distance from the axis of rotation, and in the foot flag pole example, spinning it end over end involves each extreme end 15 feet away from the axis of rotation.

However, if you twirl it around the central axis, everything is quite close to the axis. This is why you need a different equation to describe the moment of inertia for the same object depending on the rotation axis. The axis you choose affects how far parts of the body are from the axis of rotation, even though the mass of the body remains the same. The key to calculating the moment of inertia for a rigid body is learning to use and apply the appropriate equations.

Consider the pencil from the previous section, being spun end-over-end around a central point along its length. So modeling the object as a rod, you would use the following equation to find the moment of inertia, combined with the total mass and length of the pencil:. For example, consider two balls connected together by a rod which we will treat as massless to simplify the problem.

Ball one is 2 kg and positioned 2 m away from the axis of rotation, and ball two is 5 kg in mass and 3 m away from the rotation axis. In this case, you can find the moment of inertia for this composite object by considering each ball to be a point mass and working from the basic definition that:. With the subscripts simply differentiating between different objects i. The two-ball object would then have:.

Angular momentum the rotational analogue for linear momentum is defined as the product of the rotational inertia i. The equation for angular momentum L is:. Consider an ice skater spinning at a constant angular velocity with arms outstretched, and note that his arms being outstretched increases the radius R about which his mass is distributed, leading to a greater moment of inertia than if his arms were close to his body.

If L 1 is calculated with his arms outstretched, and L 2after drawing his arms in must have the same value because angular momentum is conservedwhat happens if he decreases his moment of inertia by drawing in his arms? By outstretching their legs and tail, they increase their moment of inertia and reduce the speed of their rotation, and conversely they can draw in their legs to decrease their moment of inertia and increase their speed of rotation.

Continuing the parallels between linear motion and rotational motion, objects also have rotational kinetic energy in the same way they have linear kinetic energy. Think about a ball rolling across the ground, both rotating about its central axis and moving forward in a linear fashion: The total kinetic energy of the ball is the sum of its linear kinetic energy E k and its rotational kinetic energy E rot.

You can clearly see that both equations have exactly the same form, with the appropriate rotational analogues substituted for the rotational kinetic energy equation. Considering the ball, and modeling the object as a solid sphere, the equation is this case is:.

Depending on the situation, an object might possess only linear kinetic energy for example, a ball dropped from a height with no spin imparted onto it or only rotational kinetic energy a ball spinning but staying in place. Remember that it is total energy that is conserved.

Lee Johnson is a freelance writer and science enthusiast, with a passion for distilling complex concepts into simple, digestible language. He's written about science for several websites including eHow UK and WiseGeek, mainly covering physics and astronomy. He was also a science blogger for Elements Behavioral Health's blog network for five years.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. 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. Putting this all together, we obtain.

The last step is to be careful about our limits of integration. This gives us. Next, we calculate the moment of inertia for the same uniform thin rod but with a different axis choice so we can compare the results. We would expect the moment of inertia to be smaller about an axis through the center of mass than the endpoint axis, just as it was for the barbell example at the start of this section. This happens because more mass is distributed farther from the axis of rotation. Just as before, we obtain.

However, this time we have different limits of integration. Therefore we find.A mass m is placed on a rod of length r and negligible mass, and constrained to rotate about a fixed axis. If the mass is released from a horizontal orientation, it can be described either in terms of force and accleration with Newton's second law for linear motion, or as a pure rotation about the axis with Newton's second law for rotation.

This provides a setting for comparing linear and rotational quantities for the same system. This process leads to the expression for the moment of inertia of a point mass. For a uniform rod with negligible thickness, the moment of inertia about its center of mass is. The moment of inertia about the end of the rod can be calculated directly or obtained from the center of mass expression by use of the Parallel axis theorem.

If the thickness is not negligible, then the expression for I of a cylinder about its end can be used. Calculating the moment of inertia of a rod about its center of mass is a good example of the need for calculus to deal with the properties of continuous mass distributions.

The resulting infinite sum is called an integral. The general form for the moment of inertia is:. When the mass element dm is expressed in terms of a length element dr along the rod and the sum taken over the entire length, the integral takes the form:. The moment of inertia calculation for a uniform rod involves expressing any mass element in terms of a distance element dr along the rod. To perform the integral, it is necessary to express eveything in the integral in terms of one variable, in this case the length variable r.

The integral is of polynomial type:. Once the moment of inertia of an object about its center of mass has been determined, the moment about any other axis can be determined by the Parallel axis theorem :.

Rotational and Linear Example A mass m is placed on a rod of length r and negligible mass, and constrained to rotate about a fixed axis. Index Moment of inertia concepts Moment of inertia examples. Moment of Inertia: Rod For a uniform rod with negligible thickness, the moment of inertia about its center of mass is. Index Moment of inertia concepts. Moment of Inertia: Rod Calculating the moment of inertia of a rod about its center of mass is a good example of the need for calculus to deal with the properties of continuous mass distributions.

The general form for the moment of inertia is: When the mass element dm is expressed in terms of a length element dr along the rod and the sum taken over the entire length, the integral takes the form:. Rod Moment Calculation The moment of inertia calculation for a uniform rod involves expressing any mass element in terms of a distance element dr along the rod.

Rod Moment About End Once the moment of inertia of an object about its center of mass has been determined, the moment about any other axis can be determined by the Parallel axis theorem :.Which detail from Heart of Darkness shows the ineffectiveness of the colonizers. What is the answer to this logical question pumara ako sumakay ako umupo ako sumandal ako bumaba ako anong dala ko? All Rights Reserved.

The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. Hottest Questions. Previously Viewed. Unanswered Questions. Wiki User This would depend on what axis the rod is rotating about.

Related Questions Asked in Physics What happens to the moment of inertia of a thin rod when it s length is doubled? Since its length is doubled, the number of molecules present increase, resulting in an increase in mass of the rod. And we all know that mass is directly proportional to inertia, therefore the moment of inertia also increases. Asked in Physics What happens to moment of inertia of thin rod when its length is doubled?

Asked in Physics How do you determine the moment of inertia? How do to determine the moment of inertia of a cub. Asked in Physics Mass moment of inertia of a disk? The moment of inertia is the measure of resistance to torque applied on a spinning object i. Asked in Physics, Mechanical Engineering What is the domensions for moment of inertia? Asked in Mechanical Engineering What is meant by moment of inertia in Induction motors? The second moment of a force is called as moment of inertia.

Asked in The Difference Between, Rotation What is the difference between mass moment of inertia and area moment of inertia? Asked in Science, Physics What is the unit for a polar moment of inertia? Moment of inertia has unit kg m2. Asked in Physics, Mechanics Importance of moment of inertia?

The moment of inertia tells us how easy or how difficult it is to make something rotate. Asked in Physics, Mechanical Engineering What is geometrical moment of inertia? There are two types of moments of inertia - one is mass moment of inertia and the other geometrical or area moment of inertia.

The geometrical moment of inertia is cross sectional property of an area that is used to compute stress and deflection in structures. It is the the integral of square of the distance to the centroid axis over the area cross section. Asked in Physics What is physical significance of moment of inertia? Asked in Physics Must an object be rotating to have a nonzero moment of inertia? Asked in Rotation Moment of inertia of hollow cylinder?

Here I is inertia, M is mass and R is the radius. Asked in Physics, The Difference Between What is difference between polar moment of inertia and moment of inertia? Polar moment of inertia of an area is a quantity used to predict an object's ability to resist torsion.

Moment of inertia, also called mass moment of inertia or the angular mass, SI units kg m2, Imperial Unit slug ft2 is a measure of an object's resistance to changes in its rotation rate. Asked in Physics, Isaac Newton Why moment of inertia is also called rotational inertia?

Because it is a measure of the "resistence" of an object to be accelerated in its rotation.Why don't fictional characters say "goodbye" when they hang up a phone?

If we can't tunnel through the Earth, how do we know what's at its center? All Rights Reserved. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply.

Hottest Questions. Previously Viewed. Unanswered Questions.

Wiki User This would depend on what axis the rod is rotating about. Related Questions Asked in Physics What happens to the moment of inertia of a thin rod when it s length is doubled? Since its length is doubled, the number of molecules present increase, resulting in an increase in mass of the rod.

And we all know that mass is directly proportional to inertia, therefore the moment of inertia also increases. Asked in Physics What happens to moment of inertia of thin rod when its length is doubled?

## Moment of Inertia (Angular & Rotational Inertia): Definition, Equation, Units

Asked in Physics How do you determine the moment of inertia? How do to determine the moment of inertia of a cub. Asked in Physics Mass moment of inertia of a disk? The moment of inertia is the measure of resistance to torque applied on a spinning object i.

Asked in Physics, Mechanical Engineering What is the domensions for moment of inertia? Asked in Mechanical Engineering What is meant by moment of inertia in Induction motors? The second moment of a force is called as moment of inertia. Asked in The Difference Between, Rotation What is the difference between mass moment of inertia and area moment of inertia?

Asked in Science, Physics What is the unit for a polar moment of inertia? Moment of inertia has unit kg m2. Asked in Physics, Mechanics Importance of moment of inertia? The moment of inertia tells us how easy or how difficult it is to make something rotate.

Asked in Physics, Mechanical Engineering What is geometrical moment of inertia? There are two types of moments of inertia - one is mass moment of inertia and the other geometrical or area moment of inertia. The geometrical moment of inertia is cross sectional property of an area that is used to compute stress and deflection in structures.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:.

## Ask a Question

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. 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. 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 xas shown in the figure. Putting this all together, we obtain.

The last step is to be careful about our limits of integration. This gives us.We can obtain a general idea of the rotation of rigid bodies and compare it with real situations or we can consider some real bodies as nearly rigid to obtain our results. The necessity of deriving an expression of kinetic energy of a rotating rigid body before defining moment of inertia lies inside the idea of defining moment of inertia. It means we first find the kinetic energy of a rotating rigid body and compare it with the kinetic energy of a particle in linear motion - where we can find the measure of mass of a rotating body called moment of inertia.

Consider a rigid body which rotates about y-axis as shown in Figure 1. When the body rotates, each particle of the body moves in its own circle of a particular radius centred on the axis. That's because a particle at larger distance from the axis of rotation of the body needs to move along a circle of larger radius but the time is the same for all particles to complete one rotation that is, the angular velocity for all particles is constant. So the total kinetic energy of the rotating rigid body is the sum of the kinetic energies of all particles rotating in the body.

So the above expression becomes. The kinetic energy of a rotating rigid body about an axis is defined in terms of the moment of inertia and the angular speed. In words you can define the moment of inertia of a body rotating about an axis as the product of its mass and square of its distance from the axis of rotation.

If a body is not only a point mass but a rigid body of uniform mass distribution, you can not simply take the body as a single point mass but instead you have to define a particular element of the body and use the integration to find the moment of inertia of the whole body. In fact you can not tell exactly how much moment of inertia a particular body has. It means you can make as many axes as possible about which a rigid body can rotate. So the moment of inertia of the same body is different for different rotation axis.

That's the reason why you have to tell about which axis the rigid body rotates. Here we discuss about the moment of inertia of some uniform mass distributions. Moment of inertia is not constant for non-rigid bodies even if the rotation axis is the same and it is because the distance of particles of non-rigid body from the axis of rotation varies.

So you can write. Here we find the moment of inertia of a cylinder. You can also find the moment of inertia of a disk in fact a disk is a cylinder.

We'll use the moment of inertia of a disk to find the moment of inertia of a solid sphere later. The cylinder rotates about an axis which passes through the centres of the cross sections of the cylinder. So the moment of inertia of the element is. This is also the moment of inertia of a disk. This result is the same as the moment of inertia of a thin circular ring about the axis through its centre and perpendicular to its plane. Consider a spherical shell which rotates about the x-axis with its centre at the origin of our coordinate system as shown in Figure 4.

The ratio of the mass of the element to the total mass of the spherical shell is equal to the ratio of the area of the element to the total area of the shell that is. The moment of inertia of the element about the axis in our case x-axis through its centre perpendicular to its plane is the same as the moment of inertia of a thin ring about the axis through its centre perpendicular to its plane. Check the previous discussion for the moment of inertia of a cylinder.

Therefore, the small moment of inertia of the element is. The total moment of inertia of the whole spherical shell is two times the moment of inertia of the half of the spherical shell. So the total moment of inertia is.

Physics - Mechanics: Moment of Inertia (3 of 6) Derivation of Moment of Inertia of a Bar

The centre of the sphere is the origin of our coordinate system. The ratio of the mass of the disk element to the total mass of the sphere is equal to the ratio of the volume of the disk element to the total volume of the sphere. As already discussed, the moment of inertia of a disk about the axis through its centre perpendicular to its plane is the same as the moment of inertia of a solid cylinder about the corresponding axis. Therefore, the moment of inertia of the disk element is.

So the total moment of inertia of the sphere about the axis along its diameter is twice of this:. Figure 1 A rigid body rotates about y-axis.