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Let us consider the small mass of volume "dV" of the sphere of thickness "dr", which is situated at a linear distance "r" from the center of the sphere.

(ii) Elemental mass :

Volumetric density, ρ, is the appropriate density type in this case.

ρ = M V

where "M" and "V" are the mass and volume of the uniform solid sphere respectively. Here, hollow sphere of infinitesimal thickness itself is considered as elemental mass (dm) :

đ m = ρ đ V = ( M 4 3 π R 3 ) 4 π r 2 đ r

(iii) Moment of inertia for elemental mass

Moment of inertia of elemental mass is obtained by using expresion of MI of hollow sphere :

đ I = 2 3 r 2 đ m

đ I = 2 3 r 2 ( 3 M R 3 ) r 2 đ r = ( 2 M R 3 ) r 4 đ r

(iv) Moment of inertia of rigid body

I = ( 2 M R 3 ) r 4 đ r

Taking out the constants from the integral sign, we have :

I = ( 2 M R 3 ) r 4 đ r

The appropriate limits of integral in this case are 0 and R. Hence,

I = ( 2 M R 3 ) 0 R r 4 đ r

I = ( 2 M R 3 ) [ r 4 4 ] 0 R = 2 5 M R 2

Radius of gyration (k)

An inspection of the expressions of MIs of different bodies reveals that it is directly proportional to mass of the body. It means that a heavier body will require greater torque to initiate rotation or to change angular velocity of rotating body. For this reason, the wheel of the railway car is made heavy so that it is easier to maintain speed on the track.

Further, MI is directly proportional to the square of the radius of circular object (objects having radius). This indicates that geometric dimensions of the body have profound effect on MI and thereby on rotational inertia of the body to external torque. Engineers can take advantage of this fact as they can design rotating part of a given mass to have different MIs by appropriately distributing mass either closer to the axis or away from it.

The MIs of ring, disk, hollow cylinder, solid cylinder, hollow sphere and solid sphere about their central axes are M R 2 , M R 2 2 , M R 2 , 2 M R 2 3 and 2 M R 2 5 respectively. Among these, the MIs of a ring and hollow cylinder are M R 2 as all mass elements are distributed at equal distance from the central axis. For other geometric bodies that we have considered, we find thaheir MIs expressions involve some fraction of M R 2 as mass distribution is not equidistant from the central axis.

We can, however, assume each of other geometric bodies (not necessarily involving radius) equivalent to a ring (i.e a hoop) of same mass and a radius known as "radius of gyration". In that case, MI of a rigid body can be written in terms of an equivalent ring as :

I = M K 2

Formally, we can define radius of gyration as the radius of an equivalent ring of same moment of inertia. In the case of solid sphere, the MI about one of the diameters is :

I = 2 M R 2 5

By comparison, the radius of gyration of solid sphere about its diameter is :

K = ( 2 5 ) x R

Evidently, radius of gyration of a rigid body is specific to a given axis of rotation. For example, MI of solid sphere about an axis parallel to its diameter and touching its surface (we shall determine MI about parallel axis in the next module) is :

I = 7 M R 2 5

Thus, radius of gyration of solid sphere about this parallel axis is :

K = ( 7 5 ) x R

The radius of gyration is a general concept for rotational motion of a rigid body like moment of inertia and is not limited to circularly shaped bodies, involving radius. For example, radius of gyration of a rectangular plate about a perpendicular axis passing through its COM and parallel to one of its breadth (length “a” and breadth “b”) is :

I = M a 2 12

and the radius of gyration about the axis is :

K = a 12


1. The results of some of the known geometric bodies about their central axes, as derived in this module, are given in the figures below :

Moment of inertia of known geometric bodies

Moment of inertia of rod, plate, ring and hollow cylinders
Moment of inertia of circular disk, solid cylinder, hollow sphere and solid sphere

2. The radius of gyration (K) is defined as the radius of an equivalent ring of same moment of inertia. The radius of gyration is defined for a given axis of rotation and has the unit that of length. The moment of inertia of a rigid body about axis of rotation, in terms of radius of gyration, is expressed as :

I = M K 2

3. Some other MIs of interesting bodies are given here. These results can be derived as well in the same fashion with suitably applying the limits of integration.

(i)The MI of a thick walled cylinder ( R 1 and R 2 are the inner and outer radii)

I = M ( R 1 2 + R 2 2 ) 2

(ii) The MI of a thick walled hollow sphere ( R 1 and R 2 are the inner and outer radii )

I = 2 5 x M x ( R 2 5 - R 1 5 ) ( R 2 3 - R 1 3 )

Questions & Answers

what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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can anyone tell who founded equations of motion !?
Ztechy Reply
n=a+b/T² find the linear express
Donsmart Reply
Sultan Reply
Moment of inertia of a bar in terms of perpendicular axis theorem
Sultan Reply
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Yara Reply
Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
Rama Reply
A balloon is released from the ground which rises vertically up with acceleration 1.4m/sec^2.a ball is released from the balloon 20 second after the balloon has left the ground. The maximum height reached by the ball from the ground is
Lucky Reply
work done by frictional force formula
Sudeer Reply
Misthu Reply
Why are we takingspherical surface area in case of solid sphere
Saswat Reply
In all situatuons, what can I generalize?
Cart Reply
the body travels the distance of d=( 14+- 0.2)m in t=( 4.0 +- 0.3) s calculate it's velocity with error limit find Percentage error
Clinton Reply
Explain it ?Fy=?sN?mg=0?N=mg?s
Admire Reply

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