Important Notes on Center of Mass & Moment of Inertia

By Prashant Kumar|Updated : August 3rd, 2017

 Read the following points and get the jist of the Center of Mass and Moment of Inertia in just 15 keypoints:

  1. A rigid body is the one that has a fixed shape and that cannot be unchanged. The distance between the particles is fixed and does not change.
  2. Any rigid body consists of infinite number of particles, thus called a system of particles.
  3. The whole mass of a body is said to be concentrated at a point which is called center of mass (COM)  
  4. For a single rigid body, the center of mass is usually coincides with its centroid. But the COM can lie inside the body as well as outside as in the case of hollow bodies. For example, in case of shells etc, it lies outsides the bodies.
  5. The velocity of rigid body can also be considered as the velocity of its COM.
  6. In the translation motion, the velocity of all the particles of the rigid body is same. But in rotational motion, this is not true.
  7. The line along which the object is fixed and the motion of any object is restricted is called the Axis of rotation.
  8. In the rotation about a fixed axis, every particle moves in a circle which lies in the plane perpendicular to the axis of rotation, as shown below:byjusexamprep
  9. The particle in the circular motion have angular velocity, ω, which is related to the linear velocity as v = rω.
  10. The angular acceleration, α, is the rate of change of the angular velocity with time.
  11. The rotational analogue of force, in the rotational motion is called the moment of force, also called as Torque.
  12. The torque is defined as the cross product of perpendicular distance from the axis of rotation and force. i.e. Τ=r×F.
  13. The analog of mass in rotational motion is called the Moment of Inertia or (M.I.). Also called the Angular mass or the Rotational Inertia. It is the quantity that determines the torque for angular acceleration along the axis.
  14. The unit of M.I. is kg-mand the dimensional formula is ML2.
  15. Radius of Gyration: If M is mass and І is moment of inertia of a rigid body, then the radius of gyration (k) of a body is:

    Moment of Inertia and Rigid (12)


How to find moment of Inertia:

The moment of inertia about an axis of a body is calculated by the summation of mr2 for every particle in the body, where “m” is the mass of the particle and “r” is the perpendicular distance from the axis. i.e. I = miri2

 When the distribution of mass is continuous, the discrete sum Moment of Inertia and Rigid (7) becomes Moment of Inertia and Rigid (8). For accuracy the contributions of infinitesimal mass elements dm are taken, each contributing Moment of Inertia and Rigid (9) dm to the moment of inertia. Be careful while choosing the mass element. All the particles on this element should be at same perpendicular distance from the axis. Now MOI becomes

Moment of Inertia and Rigid (10)

‘r’ is the perpendicular distance from an axis (not origin).

Moment of Inertia and Rigid (11)



Moments of Іnertia of some important bodies:

  1. Circular Ring:
    Axis passing the centre and perpendicular to the plane of ring.
    Moment of Inertia and Rigid (15)                            Moment of Inertia and Rigid (13)
  2. Hollow Cylinder:
    Moment of Inertia and Rigid (15)                                      Moment of Inertia and Rigid (14)
  3. Solid Cylinder & a Disc:
    About its geometrical axis:
    Moment of Inertia and Rigid (17)
    About its perpendicular axis:
    Moment of Inertia and Rigid (18)                          Moment of Inertia and Rigid (16)
  4. (a) Solid Sphere:
    Axis through the centre
    I = (2/5) MR2                                 Moment of Inertia and Rigid (19)

    (b) Hollow Sphere:
    Axis through the centre:
    Moment of Inertia and Rigid (22)                               Moment of Inertia and Rigid (21)
    (c) Thin Rod of length l:
    Axis through the mid – point and perpendicular to length: Moment of Inertia and Rigid (23)   Moment of Inertia and Rigid (24)
    Axis through an end and perpendicular to the rod: Moment of Inertia and Rigid (25)
  5. Regular Plate:
    Axis through centre and parallel to the height:
    Moment of Inertia and Rigid (27)                              Moment of Inertia and Rigid (28)


We hope you found these key-points useful. Just take a look at these when you revise the topic CENTER OF MASS AND ROTATIONAL MOTION. 

We will keep posting notes on other topics very soon. Keep calm and Study hard! Good luck!!


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Load Previous Comments
Atharva Kunte
2/3 mr² for hollow sphere

TanmayFeb 11, 2017

Can you tell me moment of inertia of cube rotating from its body diagonal
Adityaa Ms

Adityaa MsAug 3, 2017

Can u pls send notes in pdf format I think it would be better
Ramgopal Pandey
Some formulas are wrong
Pallavi Kochar
Moment of inertia for solid sphere is 2/5MR²

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