Moment of inertia of uniform bodies with simple geometrical shapes

Moment of inertia, often known as rotational inertia, is an essential term in Physics. It is really mr2 around a centre point of rotation, hence the unit is kgm2. It is denoted by the sign I. This is described as the resistance that a body provides to its speed of rotation or angular acceleration along an axis by applying turning force, also known as torque. Because the moment of inertia varies with the shape and size of the body, it is calculated independently in the following sections. The derivations are crucial for the numerical problems that are based on this topic.

Moment of inertia 

Definition

A rigid body’s moment of inertia, also known as its mass moment of inertia, angular mass, second moment of mass, or, more precisely, rotational inertia, is a quantity that determines the torque required for a desired angular acceleration about a rotational axis, similarly to how mass determines the force required for a desired acceleration. It is determined by the mass distribution of the body and the axis selected, with higher moments necessitating more torque to modify the body’s rate of rotation. 

An object’s moment of inertia is a calculated measurement for a rigid body revolving around a fixed axis. The axis may be internal or exterior, and it may or may not be fixed. The moment of inertia (I) is, however, always stated in reference to that axis.

The moment of inertia is affected by the distribution of mass around the axis of rotation. The MOI changes based on the axis position chosen. That is, the same item may have different moments of inertia values depending on the position and direction of the axis of rotation.

Moment of inertia

In general, Moment of Inertia is written as I = m r2.

Where m = the sum of the mass’s products

r = Distance from the rotation’s axis.

Integral form: I = ∫ r2 dm

The moment of inertia plays a very important role as the mass plays in the linear motion.

It is also a body’s resistance measurement, similar to a change in the rotational motion. 

Moment of inertia is constant for a specified rigid frame and also for the axis of rotation.

The moment of inertia, I =∑ mi ri2… (1)

Kinetic Energy, K =½ I ω2……………. (2)

Examples of Moment of Inertia

The hydrogen molecule’s moment of inertia was historically significant. It’s simple to calculate: the nuclei (protons) contain 99.95 percent of the mass, thus a traditional depiction of two point masses separated by a set distance yields I=12ma2. The problem in the nineteenth century was that equipartition of energy, which offered a good explanation of the specific heats of practically all gases, didn’t work for hydrogen—apparently, at low temperatures, these diatomic molecules didn’t spin around, while continually colliding with each other. The conclusion was that the moment of inertia was so low that it took a lot of energy to ignite the initial quantized angular momentum state, L=. This was not the case with heavier diatomic gases since their energy was higher. For molecules with larger moments of inertia, the lowest angular momentum state E=L2/2I=ℏ2/2I is lower.

Moment of inertia for different objects

The moment of inertia is impacted by the axis of rotation. Whatever we’ve discovered thus far is the moment of inertia of those items when the axis passes through their centre of mass. When you choose two unique axes, you will see that the item resists rotational change in various ways.

Moment  of inertia of sphere

The sphere’s moment of inertia is commonly represented as;

I = (⅖)MR2

R and M represent the radius and mass of the sphere, respectively. Furthermore, the moment of inertia of the sphere about its axis on the surface is represented as;

I = (7/5) MR2

Derivation

Calculate the moment of inertia of a homogeneous solid sphere about any axis passing through its centre.

I=1/2MR2

Now, 

dI=1/2r2 dm

For finding dm

dm=ρdV

For dV

dV=πr2 dx

After putting the value of dV into dm

dm=ρπr2 dx

Now substitute the value of dm into dI

dI=1/2ρπr4 dx

We must now introduce x into the equation. Above, notice how x, r, and R form a triangle. As a result of Pythagoras’ theorem.

r2=R2–x2

or, 

dI=1/2ρπ(R2–x2)dx2

Therefore,

I=1/2ρπR∫−R(R2–x2)dx2 

After integration,

I=1/2ρπ(16/15)R5

Now we must determine the density of the sphere:

ρ=MV

ρ=M/(4/3πR3)

After substituting the value

I=2/5 MR2

Conclusion

Because the moment of inertia varies with the shape and size of the body. A rigid body’s moment of inertia, also known as its mass moment of inertia, angular mass, second moment of mass, or, more precisely, rotational inertia, is a quantity that determines the torque required for a desired angular acceleration about a rotational axis, similarly to how mass determines the force required for a desired acceleration. The moment of inertia is affected by the distribution of mass around the axis of rotation. That is, the same object may have different moments of inertia values depending on the position and direction of the axis of rotation. The formula is expressed as I = m r2 where, m = Sum of the product of the mass, r = Distance from the axis of the rotation.