Moment of Inertia

Introduction

Moment of inertia is a particular topic utilized in various physics fields. It is commonly used in problems related to Mass in rotational motion and for calculating angular momentum. It plays a significant role in rotational kinematics and linear kinetics. Moment of inertia is also used in the kinetic energy, momentum and Newton’s laws of motion for a Rigid body. Rotational motion depends on the distribution of mass around the axis of rotation, which changes by changing the axis. Although, in rigid bodies, a moment of inertia is the summation of small quantities of mass multiplied by the square of the distance (distance from the axis).

Moment of Inertia Definition 

Moment of inertia is also called rotational inertia or angular mass. It can be defined as a value that decides the torque needed for an angular acceleration in the rotational axis. It is mainly considered for a chosen axis of rotation. So, it changes by the change in mass around the axis. Kg m2 is the SI unit of moment of Inertia. Although, the unit for area moment of inertia is mm4 or in4, and for mass moment of inertia is kg.m2 or ft.lb.s2. 

Moment of Inertia Formula 

The moment of inertia is the value due to the resisting angular acceleration and is the summation of the product of the mass of each particle with the distance square. 

So, in simple words, 

Moment of inertia, I = m × r2 

where,

Summation of the PTO of mass = m

Value of distance from the rotating axis = r

On integration, 

I = ∫dI = ∫M r2 dm

And the dimensional formula for the moment of inertia can be described as an M1 L2 T0. 

The mass of inertia plays the same role as the mass of linear motion. It can also be measured as the body’s resistance by changing its rotational motion. Moment of inertia remains constant for rigid frame and rotation in specific axes. 

Moment of inertia, I = ∑mi ri2

Moment of Inertia of a system of particles

The Moment of inertia of a system of particles can be described as a,

I = ∑ mi ri2

ri = ith particle perpendicular distance from the axis. 

mi = Mass of the particle 

Moment of inertia of rigid bodies 

The moment of inertia of rigid bodies can be calculated by integrating. If the system of Rigid Bodies is divided into an infinite number of particles, then its Mass, ‘dm’ and distance of Mass from the axis of rotation is ‘r’. Now, the moment of Inertia becomes, 

I = ∫ r2 dm

Moment of Inertia of different Rigid bodies

Factors that affect the Moment of Inertia

The moment of inertia is directly proportional to the mass and distance (from the axis). So, if anybody’s mass increases, the moment of inertia also increases and vice versa. Similarly, if the distance (from the axis) increases, the moment of inertia also increases and vice versa. Along with this, the moment of inertia also depends on three more factors;

• The density of the rigid body
• Shape and size of the rigid body 
• Mass distribution relative to the axis or axis of rotation. 

If any of the factors changes, the moment of inertia also changes. 

Applications of Moment of Inertia 

• Moment of inertia plays a significant role in rotational dynamics and statistics. It is also used in essential dynamics and linear statistics. 
• Moment of inertia is also used to derive and calculate torque equations. In this, it also helps in analysing the dynamics of anybody.

It can be described as a,      

       τ=Iα

Where 

τ is the torque of the rigid body. 

α is the angular acceleration of the rigid body. 

I is the moment of inertia of the rigid body. 

• Moment of inertia is used in the sugarcane crushing machine. If the wheel with a high moment of inertia is used, then the angular acceleration increases and the sugarcane crumbles easily and fastly. 
• In bicycles, long pedals are used for increasing the mass of inertia. As the mass of inertia increases, the angular acceleration also increases, due to which the wheel of bicycles rotates several times, even on less force. 
• Same in the spanner. The spanner uses the long handle to reduce the amount of force required to loosen the nut. 

Conclusion

Moment of inertia is the study of rotating motion and angular motion. Along with the distance and mass of the body, the torque on the body also impacts the moment of inertia. The moment of inertia is used in the various fields of physics, like dynamics and in maths like statistics. It also consists of different applications; some of them are written above.