Moment of Inertia & Mass Distribution - Online Tutor, Practice Problems & Exam Prep

The moment of inertia (I) of a system is influenced by the distribution of mass around the axis of rotation. For point masses, the equation is $m^2+r^2$, where r is the distance from the axis. Greater distances from the axis result in higher moments of inertia. Thus, a configuration with masses farther from the center exhibits the greatest inertia, affecting rotational speed under applied force. Understanding this concept is crucial for analyzing rotational dynamics.

Moment of Inertia & Mass Distribution

Hey, guys. So in this quick video, I'm going to show you how the moment of inertia of a system has to do with how the mass is distributed, how the masses are spread out, around the axis of rotation. Let's check it out. It says here the moment of inertia, I, has to do with how mass is distributed, how it's spread out, around an axis of rotation. So here we have a solid disk that has small masses. This is the disk and the masses are the black dots, the 4 black dots. And they're arranged in 3, not 2, 3 different ways. And I want to know in which of these, will the moment of inertia be greater? In which of these will the moment of inertia be greater?

Now this is a composite system with a bunch of different masses. So the total moment of inertia of the system would be the moment of inertia of the solid disk, which is a solid cylinder, plus the moment of inertia of the 4 masses. Okay? So something like i1 + i2 + i3 + i4. Now these three situations have the same disk with the same mass with the same radius. So for all of them, this is going to be the same. The only thing that will change is this. So the difference will be in how the tiny masses are arranged around the disk.

Now, if these are point masses, which they should be treated as point masses because it says here small mass, the equation for them is mr2. So you have a bunch of mr2, right? Four times. Now if you have the same four masses everywhere, these m's will also be the same. So it's going to come down to the r's for each mass. In other words, how far from the axis of rotation they are. Okay. So basically, the farther the masses are, the greater their individual moments of inertia will be and the greater the total moment of inertia of the system will be. So this one has to be the one with the greatest I. Okay? So I'm going to call this a, b, and c. And b is the greater one.

Now and that's because the masses are farther out from the center. C is the smallest, the lowest value of I because the masses are congregated in the middle. Here, you can see 4 masses really close to the center. Here, you'll see 4 masses really far from the center. And this guy is somewhere in the middle. 2 are far and 2 are close. So I'm going to say that the moment of inertia of b is the greater and the moment of inertia of c is the smallest. Okay? So greatest, smallest, and a is in the middle. This means that you can think of b as being the heaviest of the 3. Okay? Even if the masses are the same, it's got the most inertia. Another way that this question could be asked is, you know, if you apply the same force to it, who's going to rotate faster? Right? Well, this guy is the heaviest, so it's also going to be the slowest. Okay? Alright. So that's it for this one. Let's keep going.

The objects below all have the same mass and radius. Mass is distributed evenly in all objects. Rank the objects according to the Moment of Inertia they each have about a central axis perpendicular to them, highest to lowest. (From left to right, the objects are A, B, C, and D.)

• A. I_B > I_A > I_C > I_D
• B. I_B > I_C > I_A > I_D
• C. I_D > I_A > I_C > I_B
• D. I_D > I_C > I_A > I_B

Here’s what students ask on this topic:

What is the moment of inertia and how is it related to mass distribution?

The moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion. It depends on the mass of the object and how that mass is distributed relative to the axis of rotation. For point masses, the moment of inertia is calculated using the formula $I_i=m_i{r}^2$, where $m_i$ is the mass and $r_i$ is the distance from the axis. The farther the mass is from the axis, the greater the moment of inertia. This concept is crucial for understanding rotational dynamics and how different mass distributions affect rotational speed and stability.

How do you calculate the moment of inertia for a composite system?

To calculate the moment of inertia for a composite system, you sum the moments of inertia of all individual components. For example, if you have a solid disk with small point masses attached, the total moment of inertia (I_total) is the sum of the moment of inertia of the disk (I_disk) and the moments of inertia of the point masses (I₁, I₂, I₃, I₄). Mathematically, this is expressed as $I$_total = I_disk + I₁ + I₂ + I₃ + I₄. Each point mass's moment of inertia is calculated using $I=m{r}^2$, where m is the mass and r is the distance from the axis of rotation.

Why does the moment of inertia increase with the distance of mass from the axis of rotation?

The moment of inertia increases with the distance of mass from the axis of rotation because it is proportional to the square of that distance. The formula for the moment of inertia of a point mass is $I=m{r}^2$, where m is the mass and r is the distance from the axis. As r increases, $r^2$ increases much faster, leading to a higher moment of inertia. This means that masses farther from the axis contribute more to the system's resistance to rotational changes, making it harder to start or stop spinning.

How does the distribution of mass affect the rotational speed of an object?

The distribution of mass affects the rotational speed of an object because it influences the moment of inertia. A higher moment of inertia means the object is more resistant to changes in its rotational motion. If you apply the same torque to two objects with different mass distributions, the one with the mass farther from the axis (higher moment of inertia) will rotate more slowly. Conversely, an object with mass closer to the axis (lower moment of inertia) will rotate faster. This principle is crucial in designing rotating systems, such as flywheels and gears, to achieve desired rotational speeds and stability.

What is the significance of the moment of inertia in rotational dynamics?

The moment of inertia is significant in rotational dynamics because it quantifies an object's resistance to changes in its rotational motion. It plays a role analogous to mass in linear motion. A higher moment of inertia means more torque is required to achieve the same angular acceleration. This concept is essential for understanding how different mass distributions affect the rotational behavior of objects, such as how quickly they can start or stop spinning and how stable they are during rotation. Engineers and physicists use the moment of inertia to design and analyze rotating systems, ensuring they perform as intended under various forces and conditions.