Values of Moments of Inertia

This article will discuss the moment of inertia formula, factors on which moments of inertia depend, its different types, and the importance of the moment of inertia. The moment of inertia of an object presents a continuous distribution of mass at a varying distance. Inertia accelerates with an increase in the mass; therefore, the force required to produce the linear acceleration increases.

Moments of Inertia 

• Moments of inertia measure a body’s resistance that causes its rotational motion to change. 
• Moments of inertia represent the rotational inertia and the rotational analogue of mass for linear motion. 
• Moment of inertia is specified concerning a chosen axis of rotation. 
• The calculation of moments of inertia involves calculus because, from any rotation axis, the moment of inertia of an ordinary object involves a continuous distribution of mass at a continually varying distance. 

Importance of moment of inertia 

The moment of inertia’s value is the same as the mass in a translational motion.  When we measure the inertia in a translational motion, the mass of the body is used. Thus, we can observe that the inertia increases in number with an increase in the mass, and therefore, the force required to produce the linear acceleration also increases. 

Moment of inertia formula 

The formula to calculate the moment of inertia is 

I = m r2

where m is the sum of the product of the masses and r refers to the distance from the axis of rotation. Therefore, the moment of inertia for a point mass is the product of the body’s mass of the body with the square of the perpendicular distance to the rotation axis. 

The integral representation of the moment of inertia formula is I = 0Mr2 dm.

Solved Example 1- There is a disc of radius 50 cm and mass 3 kg about an axis that passes through the center and is perpendicular to the plane of the disc. Find the moment of inertia. 

Solution- From the above statement, we can gather the following information- 

M (mass) = 3 kg

R (radius of the disc) = 50 cm = 50 10-2 = 0.5 m 

We know that the moment of inertia about an axis that passes through the center and is perpendicular to the plane of the disc is – 

I (inertia) = 12 MR2

I = 12 3 (0.5)2 

I = (0.5)3 3 

I = 0.125 3 

I = 0.375 kg m2

Therefore, the moment of inertia about an axis that passes through the center and is perpendicular to the plane of the disc is 0.375 kg m2. 

Solved Example 2- There is a disc of radius 70 cm and mass 8 kg about an axis that touches the edge and is perpendicular to the plane of the disc. Find the moment of inertia. 

Solution- From the above statement, we can gather the following information- 

M (mass) = 8 kg

R (radius of the disc) = 70 cm = 70 10-2 = 0.7 m 

We know that the moment of inertia about an axis that touches the edge and is perpendicular to the plane of the disc is I c + Md2. 

Where I c = 12 MR2 and d = R (radius of the disc). 

Therefore, I = I c + Md2 = 12 MR2 + MR2

I = 32 MR2

I = 32 8 × 0.72

I = 32 8 × 0.49

I = 5.88 kg m2

Therefore, the moment of inertia about an axis that touches the edge and is perpendicular to the plane of the disc is 5.88 kg m2. 

Conclusion 

Moment of inertia is a calculation of the required force to rotate an object. The value can be manipulated to either increase or decrease the inertia. In sports such as ice skating, diving and gymnastics, athletes constantly change their body configuration.

 The sole aim of this module is to impart knowledge about the properties of inertia, about the importance of the moment of inertia along with the moment of inertia formula and the role inertia plays in our day to day life.