Moment of Inertia of Semicircle

The syllabus of the IIT JEE for the subject of Physics covers all the chapters of two previous classes, that is to say, class 11 and class 12. And hence, it is very important for the students to study all those units. And one of those units is the System of Particles and Rotational motion, which you may have studied in class 11th Physics. In which you have learned the motion of the extended body, that is to say, the system of particles. And one of the important parts of the system of Particles is that of Moment of Inertia in Semicircle.

If you have forgotten the part of the Moment of Inertia in Semicircle, then you do not have to worry at all. Vedantu provides the complete explanation of the Moment of Inertia in Semicircle for the students of IIT JEE, and that too is free of cost.

Moment of Inertia of Semicircle

The moment of inertia plays the same role in rotational motion as the mass does in the translational motion. In other words, the moment of inertia is the measurement of 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 the rotational motion. Similarly, the larger the moment of inertia of the body, the more difficult it is to stop its rotational motion.

The moment of inertia of the semicircle is generally expressed as I = πr4 / 4

The role that the mass plays in the translation motion, is played by the moment of inertia in the rotational motion. As mentioned above the equation for the moment of inertia is \[ I = \sum m_{i} r_{i}^{2}\] but for the moment of inertia in semicircle I = πr4 / 4

For finding the moment of inertia in a semicircle, it is necessary to find the moment of inertia in a full circle first. Because a semicircle is nothing but the half of the full circle, and hence for finding the moment of inertia in the semicircle, all you have to do is to divide the moment of inertia in a full circle in half, that is to say, by two.

Also, the sum of both the axis, that is to say, the x-axis and the Y-axis are to be taken for determining the moment of inertia in a semi-circle.

The law of the moment of inertia and the moment of inertia in a semicircle is very much important in our practical life. Many machines produce rotational motion such as the Steam engine, automobile engine etc. These machines produce rotational motion because they have discs called Flywheel, which have a large amount of moment of inertia. Therefore, the moment of inertia helps in calculating the angular motion of these discs.

Finding Moment of Inertia of Semicircle 

The moment of inertia of the semicircle is generally expressed as I = πr4 / 4. Here in order to find the value of the moment of inertia of a semicircle, we have to first derive the results of the moment of inertia full circle and basically divide it by two to get the required result of that moment of inertia for a semicircle. Further to determine the moment of inertia of the semi-circle, we will take the sum of both the x and y-axis.

We know that for a  full circle because of complete symmetry and uniform area distribution, the moment of inertia relative to the x-axis is equal to that of the y-axis.

Thus, Ix = Iy = ¼ πr4

M.O.I relative to the origin, Jo = Ix + Iy = ¼ πr4 + ¼ πr4 = ½ πr4

Now we need to pull out the area of a circle which gives us:

Jo = ½ (πr2) R2

Thus similarly for the semicircle, the moment of inertia of the x-axis is equal to that of the y-axis. Here, the semi-circle rotating about an axis is symmetric, and therefore, we consider these values equal. Thus, M.O.I will be half the moment of inertia of that of a full circle. Now this gives us:

Ix = Iy = ⅛ πr4 = ⅛ (Ao) R2 = ⅛ πr2) R2

Now to find the moment of inertia of the semicircle we will take the sum of both the x and y-axis.

M.O.I relative to the origin, Jo = Ix + Iy = ⅛ πr4 + ⅛ πr4 = ¼ πr4

Concept of Area Density(σ) 

Selecting and defining a tiny strip of mass with differential width. Now writing an expression for the area density for the whole semicircle and then the tiny strips of differential widths. Add all of the individual strips using integral Calculus.

Use the concept for area density, which is mass divided by area. Area density (σ) is an intensive property, meaning that it does not 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 semicircle or a small strip of differential width.

The macro-scale area density is given in this equation: dm/da= σ

Derivation of Formula for Moment of Inertia of Semicircle

1. Here in order to derive the moment of inertia of a semicircle we define the coordinates using the polar system. We get:

z = r sin θ

y = r cos θ

2. Now we have to determine the differential area by finding the area of the element. It is given as:

ABCD is a sector with area = (r⋅d θ) ⋅ dr = r ⋅ drd θ

The centroid of this elemental area from x-axis = y sin θ

3. Using this we can find the first moment of inertia about the x-axis. We get:

\[ I_{x} = \int_{0}^{\pi} \int_{0}^{R} y . dA\] 

= \[ \int_{0}^{\pi} \int_{0}^{R} r sinθ . rdrdθ \]

= \[\int_{0}^{\pi} sin θ (\int_{0}^{R} r^{2}dr ) dθ \]

=\[ \int_{0}^{\pi} \frac{R^{3}}{3} sin θ d θ\]

= \[ \frac {R^{3}} {3} {cos θ _{0}^{\pi}}\]

= \[ \frac{2R^{3}}{3}\]

Meanwhile, Ix = A = π / 2 R2 y

We get,

π / 2 R2 y = 2R3 / 3

y = 4R / 3π

Now in order to find the second moment of inertia:

\[ I_{xx} = \int_{0}^{\pi}\int_{0}^{R}  y^{2} dA\]

\[ I_{xx} = \int_{o}^{R} sin^{2} θ ( \int_{0}^{R} r^{3} dr ) dθ \] 

\[ I_{xx} = \int_{}^{} sin^{2} θ (\frac{r^{4}}{4} )_{0}^{r} d θ \]    

\[ I_{xx} = \frac{r^4} {4}\int_{0}^{\pi}  sin^2 dθ \] 

Using the trigonometric identity: \[ sin^2θ = \frac{1-cos^2θ } {2}\] we calculate the integral.

\[ I_{xx}  = \frac{r^4}{8} {\int_{0}^{\pi}\frac{(1 – cos 2 θ ) }{2} d θ}\]   

\[ I_{xx} =\frac{( r^{4} θ– \frac{sin2θ}{2})_{0}^{\pi})}{4}\]

\[ I_{xx} = \frac{πr^{4}}{8} \]

Benefits of having the Explanation of the Moment of Inertia in a Semicircle from the Vedantu

The subject of science requires the students a great understanding of the topic. But since science is a practical subject, its theory sometimes becomes rather difficult to understand for the students of the IIT JEE. And also the science subject has a different set of vocabulary as well, and the same goes for the IIT JEE topic moment of inertia in a semi-circle.

Thereby, to understand the topic in a better manner the students must have a simple and understandable explanation of the topic so that students can score good marks in the exam. And hence, Vedantu provides the explanation in a simple and lucid language for the topic of the moment of inertia in a semicircle.

The main features of the explanation provided by Vedantu are as under:

• Simple Explanations: Vedantu provides the complete explanation of the moment of inertia in a semicircle in a simple and accessible language. So that the students do not have to face any trouble in understanding the concept.

• Provided by the Expert: one cannot explain the topic of the moment of inertia in a semicircle is a simple language if one is not an expert on the subject. That is why the explanation that Vedantu provides is provided by the expert educators who have years of experience in teaching the students the topic of Law of inertia in a semicircle.

A Brief Overview of the Moment of Inertia

The product of the mass of the object and its Velocity is the Vector quantity, known as Linear Momentum. Now, the rotational counterpart of this Linear Momentum is called “Angular Momentum”. And the Moment of Inertia is the quantity, to find out the torque, that is to say, the moment of force needed for the desired angular momentum. In the calculation of the angular Momentum, the Moment of Inertia is applied. The moment of inertia is also sometimes termed as rotational inertia or the angular mass, as well.

For the rigid body, the moment of inertia depends on quite many factors, such as the mass of the rigid body, the shape of a rigid body, distribution of the mass is both the axis of the rotation and the position and rotation of the axis of rotation.

\[ I = \sum m_{i} r_{i}^{2} \] is the equation for the moment of inertia.

If you wish to learn more about the moment of inertia you may find this link helpful:  Moment of Inertia – Explanation, Formulas, Questions with Answer and Application