Calculation of Moment of Inertia

There is an object that is independent to rotate around its axis. In this case, it is the impetus or the force of the torque that makes an object rotate. In common words, torque can be distinguished as the estimation of the force that can emerge in an object rotating about an axis. 

Just as an impetus is what prompts an object to quicken from its original position, the action of torque does the same for the rotational movements. So, the proportion of torque that is wanted by the body to undergo rotational movement is proportional to the moment of inertia that is present in the body. 

Moment of Inertia of solid sphere, moment of inertia of solid cylinder, moment of inertia of a ring and moment of inertia of disc are some of the examples of moment of inertia in different shapes. 

How do we define Moment of Inertia?

Moment of Inertia can be defined as the proportion or the quantity which is compelled to determine the amount of torque that is needed, to make the body go in angular momentum around the rotational axis. It is also recognized by terms like angular mass, rotational inertia, mass moment of inertia and second moment of mass. 

We all are aware that in cases of linear situations, the momentum of a body is conserved (law of conservation of momentum), so this goes similar to angular motions as well. Here, in angular acceleration, the angular momentum of a body defined as the moment of inertia × angular velocity remains fixed in a body. 

The moment of inertia units is asserted as 

1) kg m² (kilogram meter squared), which is the SI unit of Moment of Inertia 

2) lbf ft s² (pound-foot second squared), which is the part of US or imperial units. 

Different Definitions for Moment of Inertia

1) Scientifically, we distinguish the Moment of Inertia as the product of the mass of a section of the body and the squared result of the extent between the centroid and the reference axis around which the body rotates. 

2) The Moment of Inertia is conveyed with the formula I = L/ω. To keep the angular momentum constant in a rotating body, as the size of the moment of inertia goes down, the angular velocity must compulsorily increase to balance and equalize the situation and keep the angular momentum of a body constant. 

3) In the cases, when the shape of the body doesn’t change, the moment of inertia is defined as the ratio of the applied torque on a body with the angular acceleration of a body.

                           I = τ/α

4) In the case of a simple pendulum, the moment of inertia is defined in terms of the mass and distance undertaken from the pivot point or the axis.

                          I = mr²

5) There can be another case too, where the moment of inertia is defined in terms of the mass of the body and the effective radius of gyration of the body undertaken from the particular axis motion which takes place in the body. 

                          I = mk²

Formulas for Calculation of Moment of Inertia for various shapes

1) In the case of Rod about the center, the formula for moment of inertia is given as ML²/12. 

2) Moment of Inertia of a solid sphere is defined in terms of the mass and radius of the uniform spherical object. It is described by the formula 2MR²/5.

3) Moment of Inertia of disc is also defined in terms of the radius and mass of the object. It is given as I = MR²/2.

4) The most different of all is the case of a solid cylinder’s central diameter. The moment of inertia for this shape is given as (MR²/ 4) + (ML²/12). 

5) Moment of Inertia of a hoop about diameter is given by the formula same as that of the moment of inertia of disc. It is explained as MR²/2. 

Exemplary question

Question 1: Suppose three wooden stakes are positioned in the pattern of an equilateral triangle. Each of these stakes have length l and mass m. In this context, find the moment of Inertia of the structure that is established by these stakes about an axis passing through its prominent point of the masses and which is upright to the plane. 

Answer: Let’s consider FGH as the three sides of the equilateral triangle. Here, the perpendicular intersects the midpoint of line GH, at point I. From the centre of the masses, a line also connects side G, forming a 30° angle. 

So, the moment of inertia for the wooden stick GH can be shown as the formula I_GH = ML²/12. 

Now, the moment of inertia for this wooden stick GH about the asked axis can be explained through the parallel theorem. 

              I₁ = ML²/12 + ML (1/2√3)

                     I₁ = ML²/6

So, the Moment of Inertia of all the three wooden sticks is given as: 

                     I = 3(ML²/6) 

                      I = ML²/2

Question 2: If there is a solid sphere that has a radius equal to 5 m and mass equal to 6 kg, rotating about its own axis, then calculate the moment of inertia in this case of solid sphere. 

Answer: Here, we know that the Moment of Inertia of a solid sphere is defined in terms of the mass and radius of the uniform spherical object. It is described by the formula 2MR²/5.

So,                   I = 2[6×(5)²]/5

                        I = (2×150)/5

                              I = 60

Question 3: Suppose there is a disc having a mass 3 kg and radius 5 m, rotating around its own axis. Find the moment of inertia of the disc. 

Answer: We know that Moment of Inertia of disc is defined in terms of the radius and mass of the object. It is given as I = MR²/2.

So,                       I = 3×5² / 2

                                I = 75/2

                                I = 37.5

Conclusion 

In conclusion, we can interpret the term ‘moment of inertia’ as the quantity which specifies the amount of torque that is required to put a body in an angular momentum. In linear situations, as the momentum of a body is conserved (law of conservation of momentum). Likewise, angular acceleration comprises angular momentum of a body (defined as moment of inertia × angular velocity) which remains fixed in a body.