# Moment of Inertia of a Cube     ## Cube’s Moment of Inertia

Moment of inertia plays the same role in rotational motion as the mass does in translational motion. In other words, the moment of inertia is the measurement of the resistance of the body to a change in its rotational motion. For example, if the body is at rest. The larger the moment of inertia of the body, the more difficult it is to put the body into rotational motion. Similarly larger the moment of inertia of the body, the more difficult is to stop its rotational motion.

The moment of inertia of the cube is calculated using different equations depending on the location of its axis. When the axis of rotation is at the center:

• $I = (\frac{1}{6}) ma^{2} = \frac{ma^{2}}{6}$ when the axis of rotation passes through the center.

• $I = \frac{2mb^{2}}{3}$ when the axis of rotation passes through its edge.

### Concept of Area Density ($\sigma$)

Area Density can be found by selecting and defining a tiny strip of mass with differential width. Now write an expression for the area density for the whole cube and then the tiny strips of differential widths. Add all of the individual strips using integral Calculus.

From the concept of area density, which is mass divided by area, Area density ($\sigma$) is an intensive property, meaning that it doesn't depend on the amount of the material and also as long as the mass is uniform, its area density is the same whether you have chosen the entire or small strip of differential width.

The macro-scale area density is given in this equation: $\frac{dm}{da} = \sigma$

### Derivation of the Moment of Inertia of Cube

To derive the moment of inertia of a cube when its axis is passing through the center, we will assume the solid cube has mass m, height h, width w and depth d. Now the moment of inertia of the cube is similar to that of a square laminar with a side about an axis through the center. Also, we will be assuming the area density of the lamina to be ρ. We will then take the element of the lamina with cartesian coordinates x, y in the plane to be dx -dy. Thus, its mass is = $\rho~dxdy.$

To find the Moment of inertia of the body, we will use $\rho (x^{2} + y^{2}) dx dy$.

Now the next step involves integration where we integrate over the entire lamina. We obtain;

$\int_{-a/2} ^{a/2} \int_{-a/2} ^{a/2} \rho (x^{2} + y^{2}) dx dy = \frac{\rho a^{4}}{6}$.

We will have to then substitute the values for the mass of the lamina which is $\rho~=~ma^{2}$.

And we obtain, $I = \frac{ma^{2}}{6}$

Now the next case is when the axis is passing through the edge, we will understand how the derivation is carried out below.

The equation for moment of inertia is written as:

$I = \int r^{2}dm$

Now we need to find the MOI about an axis through the edge, we will take the z-axis.

Further moving forward, we will have to consider the cube to be broken down into infinitesimally small masses. We can thus assume their sizes to be dy, dx, and dz. With this, we get;

dm = $\rho$ dxdydz

Here, $\rho$ = density

If we look at the moment of inertia formula given above, we have r as well. It is the distance from the z-axis to mass dm. Let the coordinates of the mass ‘dm’ be x,y, and z). Now the distance ‘r’ will be;

$r= \sqrt{( x^{2} + y^{2})}$

$r^{2} = x^{2} + y^{2}$

Meanwhile, the value of x,y, and z will range from O to b according to the length of the edges.

We will now have to substitute the values that we have obtained so far in the moment of the inertia equation and finally carry out the integration.

$I = \int r^{2}dm$

$I = \int_{o}^{b} \int_{o}^{b} \int_{o}^{b} (x^{2} + y^{2}) \rho dxdydz$

$I = \frac{\rho 2b^{5}}{3}$

Mass of cube, $m = \rho b^{3}$.

Finally substituting it in the equation, we get;

$\frac{\rho 2mb^{2}}{3}$

### Finding the Moment of Inertia through Face Diagonal of the Cube

It is clear from the figure that the Moment of Inertia of a square plate can be given as

I=2×(Moment of Inertia of the triangular plate)

Thus, the Moment of Inertia of the triangular plate is given as

$I_{tri}~=~\frac{Mr^{2}}{6}$

$\therefore I = 2~\times I_{tri}$

$I~=2~\times~\frac{Mr^{2}}{6}$

And from the above figure $r~=~\frac{a}{\sqrt{2}}$

Now, we can consider the cube to be made up of square plates of dm mass stacked upon each other till height a.

So, the total moment of inertia of the cube about the diagonal is

$\int_{0}^{I} dI = \frac{1}{12}(\int_{0}^{M}a^{2}dm)$

$I~=~\frac{Ma^{2}}{12}$​

### Useful Tips

• Know the Syllabus Well and Buy the Required Material: Make sure to have all the study material at the earliest. Specific books are important as they help in identifying what topics are important for the exam and what isn't worth the trouble, because time is a crucial factor during preparation. Popular JEE books are carefully engineered to help students understand the pattern and extent of the testing areas. While those reference books are essential, a common misconception of students is that NCERT isn't that important, ignoring the fact that it is the book that helps in forming a strong base. Solving all examples and exercises from NCERT becomes the foundation of higher-order problems.

• Create a Study Schedule and Stick to it: If there's a precise and calculated plan you have ready, it enhances your productivity because it constantly makes you aware of upcoming deadlines. It betters your perspective as to how and what you put your time into. It is advisable to make a balanced and realistic schedule that incorporates everything you need to work on and at the same time doesn't tire you too much to demotivate you to follow it the next day. It must include a healthy amount of sleep and physical exercise that keeps you healthy and happy, which are other indispensable components of exam preparation.

• Eliminate all Distractions: It is hard to stay at home just toiling for a single exam all day and as a matter of course, students often succumb to distractions such as social media apps like Instagram or Twitter, etc. Therefore, it is important to eliminate these problem-causing elements before they become an issue. This is something every aspirant should realize, that small sacrifices might fetch them much greater results in the future. Procrastination and escapism through them might look enticing in the present but often turn into regret in the long run. These momentary distractions are not worthy enough to get in your way to your dreams of being alumni from IIT/NIT.

• Practice Problems Daily: Regular practice plays a crucial role in exams that have such a vast syllabus. It is not unusual to forget a topic if you haven't gone through it in a while so, make sure to regularly revisit old topics while completing the ongoing syllabus. Complete practice sheets and exercises from the study material available daily. Another additional tip is to practice time-bound so that you do not face a time crunch in the final exam. Do not resort to simply googling the answer to anything you're unable to solve. This idea is very convenient and inevitably tempting but to build a problem-solving temperament, it is necessary to fight with the question yourself and if you're still stuck, there's no harm in taking help from a teacher.

• Have a Good Hand at the Theory as well: In the midst of practicing abundant numerical, students often tend to undermine the theory. Having the theory on your tips helps you navigate through various methods of solving a question much more clearly and quickly. It gives you a fair idea of the question you are asking and what can be your approach towards it. Once you identify what concept the question revolves around, there's not much work left to do. Statistically speaking, a good chunk of physics and chemistry questions in JEE is theory-based. Being thorough with the theoretical part guarantees those marks.

• Do not Lose Heart if you can't Solve Questions in one go: JEE is hard and it's okay to not be able to get everything automatically at the beginning. That takes time and experience. If you're unable to grasp some things, keep practicing and don't lose hope. Thousands of students go through the same emotions you're facing.  Talk with someone older and experienced and identify your weak areas. Clear your doubts and take help in understanding those portions. Practice as much as you can and slowly start leveling up. Small improvements are what usher you towards the top of your game. Know that failure and success are two sides of the same coin. Always remember to strive for progress and not just perfection.

• Start Loving the Subjects: Once you genuinely like what you're doing, it makes the whole process a lot less stressful. This can be done by relating what you're studying to real life. Subjects like Math and Physics especially are highly applicable in daily life. Learn to look at the beauty of them and then there are just tools you can easily learn to use. That will also help you understand everything much better, which will make them interesting to work on. Once you think what you're beavering over is worthy of your time and energy, it automatically ignites the passion required to succeed.

## FAQs on Moment of Inertia of a Cube

1. What is the Moment of Inertia of a Cube?

The moment of inertia is the property of the mass of the rigid body that defines the total net torque needed for a desired or required angular acceleration about an axis of rotation. The mass moment of inertia of a solid cube (axis of rotation at the center of a face) is related to the length of its side.

2. How can we increase the Moment of Inertia of the body?

Moment of inertia is mainly the calculation of the required force to rotate an object. This value can be increased or decreased by the corresponding increase in the radius from the axis of rotation, the moment of inertia increases thus decreasing the speed of rotation.

3. On what factors does the moment of inertia of any given body depend?

The formula for the moment of Inertia depends upon m = mass. r = Distance from the axis of the rotation. Also, the dimensional formula of the moment of inertia can be given by,$M^{1}~L^{2}~T^{0}$.

4. Why is the importance of Moment of Inertia important?

Rotational inertia is important in Physics as this involves the mass in rotational motion. It is used to calculate the angular momentum which also allows us to explain (via conservation of angular momentum) how rotational motion changes when the distribution of mass changes.

5. What shape has the lowest Moment of Inertia?

For any given shape, the moment of inertia through the center of mass of that body will be the minimum, since any moment of inertia through another axis would add mr2 to the result.

6. What is the significance or importance of the radius of Gyration?

Since the mass of any rotating rigid body is considered to be distributed with respect to the axis of rotation, we have defined a new parameter known as the radius of gyration. It is that distance whose square when multiplied with the mass of that body gives us the moment of inertia of the body about that given axis.

Comment