Rotational Inertia Formula

Inertia is the quality of a body that opposes any force that seeks to move it or, if it is moving, to modify the amount or direction of its velocity. Rotational inertia defines the same thing for a rotating object. It is a scalar value that is directly proportional to the mass and radius of a rotating object. In rotational mechanics, rotational inertia functions similarly to mass in linear mechanics. It also relies on how that mass is distributed along the axis of rotation. It is used to compute angular momentum and to explain how rotational motion varies when the mass distribution changes.

Rotational Inertia Formula

The formula for rotational inertia of a rotating object is equal to the product of mass and square of the radius of its circular path. It is denoted by the symbol I. Its unit of measurement is kgm² and the dimensional formula is given by [M¹L²T⁰].

I = mr²

Where,

• I is the rotational inertia
• m is the mass of the rotating object
• r is the radius of the circular path

Sample Question

Question 1. Calculate the rotational inertia for an object of mass 20 kg and radius 4 m.

Solution:

We have,

m = 20

r = 4

Using the formula we have,

I = mr²

= 20 × 4 × 4

= 320 kg m²

Question 2. Calculate the rotational inertia for an object of mass 25 kg and radius 6 m.

Solution:

We have,

m = 25

r = 6

Using the formula we have,

I = mr²

= 25 × 6 × 6

= 900 kg m²

Question 3. Calculate the rotational inertia for an object of mass 11 kg and radius 4 m.

Solution:

We have,

m = 11

r = 4

Using the formula we have,

I = mr²

= 11 × 4 × 4

= 176 kg m²

Question 4. Calculate the mass of an object if its rotational inertia is 300 kg m² and radius is 10 m.

Solution:

We have,

I = 300

r = 10

Using the formula we have,

I = mr²

=> m = I/r²

=> m = 300/(10 × 10)

=> m = 300/100

=> m = 3 kg

Question 5. Calculate the mass of an object if its rotational inertia is 450 kg m² and the radius is 15 m.

Solution:

We have,

I = 450

r = 15

Using the formula we have,

I = mr²

=> m = I/r²

=> m = 450/(15 × 15)

=> m = 450/225

=> m = 2 kg

Question 6. Calculate the radius of an object if its rotational inertia is 540 kg m² and the mass is 15 kg.

Solution:

We have,

I = 540

r = 15

Using the formula we have,

I = mr²

=> r² = I/m

=> r² = 540/15

=> r² = 36

=> r = 6 m

Question 7. Calculate the radius of an object if its rotational inertia is 972 kg m² and the mass is 12 kg.

Solution:

We have,

I = 972

r = 12

Using the formula we have,

I = mr²

=> r² = I/m

=> r² = 972/12

=> r² = 81

=> r = 9 m